{"id":153,"date":"2026-01-12T15:55:36","date_gmt":"2026-01-12T15:55:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=153"},"modified":"2026-01-12T15:55:36","modified_gmt":"2026-01-12T15:55:36","slug":"techniques-for-differentiation-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/techniques-for-differentiation-get-stronger-answer-key\/","title":{"raw":"Techniques for Differentiation: Get Stronger Answer Key","rendered":"Techniques for Differentiation: Get Stronger Answer Key"},"content":{"raw":"

Derivatives of Trigonometric Functions<\/h2>\r\n
    \r\n \t
  1. [latex]\\frac{dy}{dx}=2x- \\sec x \\tan x[\/latex]<\/li>\r\n \t
  2. [latex]\\frac{dy}{dx}=2x \\cot x-x^2 \\csc^2 x[\/latex]<\/li>\r\n \t
  3. [latex]\\frac{dy}{dx}=\\frac{x \\sec x \\tan x- \\sec x}{x^2}[\/latex]<\/li>\r\n \t
  4. [latex]\\frac{dy}{dx}=(1- \\sin x)(1- \\sin x)- \\cos x(x+ \\cos x)[\/latex]<\/li>\r\n \t
  5. [latex]\\frac{dy}{dx}=\\frac{2 \\csc^2 x}{(1+ \\cot x)^2}[\/latex]<\/li>\r\n \t
  6. \r\n

    [latex]y=\u2212x[\/latex]<\/p>\r\n\r\n\"The<\/li>\r\n \t

  7. \r\n

    [latex]y=x+\\frac{2-3\\pi}{2}[\/latex]<\/p>\r\n\r\n\"The<\/li>\r\n \t

  8. [latex]y=\u2212x[\/latex]\r\n\r\n\"The<\/li>\r\n \t
  9. [latex]\\frac{d^2 y}{dx^2} = 3 \\cos x-x \\sin x[\/latex]<\/li>\r\n \t
  10. [latex]\\frac{d^2 y}{dx^2} = \\frac{1}{2} \\sin x[\/latex]<\/li>\r\n \t
  11. [latex]\\frac{d^2 y}{dx^2} = 2\\csc x( \\csc^2 x + \\cot^2 x)[\/latex]<\/li>\r\n \t
  12. [latex]x = \\frac{(2n+1)\\pi}{4}[\/latex], where [latex]n[\/latex] is an integer<\/li>\r\n \t
  13. [latex](\\frac{\\pi}{4},1), \\, (\\frac{3\\pi}{4},-1)[\/latex]<\/li>\r\n \t
  14. [latex]a=0, \\, b=3[\/latex]<\/li>\r\n \t
  15. [latex]y^{\\prime}=5 \\cos (x)[\/latex], increasing on [latex](0,\\frac{\\pi}{2}), \\, (\\frac{3\\pi}{2},\\frac{5\\pi}{2})[\/latex], and [latex](\\frac{7\\pi}{2},12)[\/latex]<\/li>\r\n \t
  16. [latex]\\frac{d^3 y}{dx^3} = 3 \\sin x[\/latex]<\/li>\r\n \t
  17. [latex]\\frac{d^4 y}{dx^4} = 5 \\cos x[\/latex]<\/li>\r\n \t
  18. [latex]\\frac{d^3 y}{dx^3} = 720x^7-5 \\tan (x) \\sec^3 (x)- \\tan^3 (x) \\sec (x)[\/latex]<\/li>\r\n<\/ol>\r\n

    The Chain Rule<\/h2>\r\n
      \r\n \t
    1. [latex]\\frac{dy}{dx} = 18u^2 \\cdot 7=18(7x-4)^2 \\cdot 7[\/latex]<\/li>\r\n \t
    2. [latex]\\frac{dy}{dx} = \u2212\\sin u \\cdot \\frac{-1}{8}=\u2212\\sin (\\frac{\u2212x}{8}) \\cdot \\frac{-1}{8}[\/latex]<\/li>\r\n \t
    3. [latex]\\frac{dy}{dx} = \\frac{8x-24}{2\\sqrt{4u+3}}=\\frac{4x-12}{\\sqrt{4x^2-24x+3}}[\/latex]<\/li>\r\n \t
    4. \r\n
        \r\n \t
      1. [latex]f(u) = u^3, \\, u=3x^2+1[\/latex]<\/li>\r\n \t
      2. [latex]\\frac{dy}{dx} = 18x(3x^2+1)^2[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
      3. \r\n
          \r\n \t
        1. [latex]f(u)=u^7, \\, u=\\frac{x}{7}+\\frac{7}{x}[\/latex]<\/li>\r\n \t
        2. [latex]\\frac{dy}{dx} = 7(\\frac{x}{7}+\\frac{7}{x})^6 \\cdot (\\frac{1}{7}-\\frac{7}{x^2})[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
        3. \r\n
            \r\n \t
          1. [latex]f(u)= \\csc u, \\, u=\\pi x+1[\/latex]<\/li>\r\n \t
          2. [latex]\\frac{dy}{dx} = \u2212\\pi \\csc (\\pi x+1) \\cdot \\cot (\\pi x+1)[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
          3. \r\n
              \r\n \t
            1. [latex]f(u)=-6u^{-3}, \\, u= \\sin x[\/latex]<\/li>\r\n \t
            2. [latex]\\frac{dy}{dx} = 18 \\sin^{-4} x \\cdot \\cos x[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
            3. [latex]\\frac{dy}{dx} = \\frac{4}{(5-2x)^3}[\/latex]<\/li>\r\n \t
            4. [latex]\\frac{dy}{dx} = 6(2x^3-x^2+6x+1)^2(3x^2-x+3)[\/latex]<\/li>\r\n \t
            5. [latex]\\frac{dy}{dx} = -3(\\tan x+ \\sin x)^{-4} \\cdot (\\sec^2 x+ \\cos x)[\/latex]<\/li>\r\n \t
            6. [latex]\\frac{dy}{dx} = -7 \\cos (\\cos 7x) \\cdot \\sin 7x[\/latex]<\/li>\r\n \t
            7. [latex]\\frac{dy}{dx} = -12 \\cot^2 (4x+1) \\cdot \\csc^2 (4x+1)[\/latex]<\/li>\r\n \t
            8. [latex]10[\/latex]<\/li>\r\n \t
            9. [latex]-\\frac{1}{8}[\/latex]<\/li>\r\n \t
            10. -[latex]4[\/latex]<\/li>\r\n \t
            11. -[latex]12[\/latex]<\/li>\r\n \t
            12. [latex]10\\frac{3}{4}[\/latex]<\/li>\r\n \t
            13. [latex]y=-\\frac{1}{2}x[\/latex]<\/li>\r\n \t
            14. [latex]x= \\pm \\sqrt{6}[\/latex]<\/li>\r\n \t
            15. \r\n
                \r\n \t
              1. [latex]-\\frac{200}{343}[\/latex] m\/s<\/li>\r\n \t
              2. [latex]\\frac{600}{2401} \\, \\text{m\/s}^2[\/latex]<\/li>\r\n \t
              3. The train is slowing down since velocity and acceleration have opposite signs<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
              4. \r\n
                  \r\n \t
                1. [latex]C^{\\prime}(x)=0.0003x^2-0.04x+3[\/latex]<\/li>\r\n \t
                2. [latex]\\frac{dC}{dt}=100 \\cdot (0.0003x^2-0.04x+3)[\/latex]<\/li>\r\n \t
                3. Approximately [latex]$90,300[\/latex] per week<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                4. \r\n
                    \r\n \t
                  1. [latex]\\frac{dS}{dt}=-\\frac{8\\pi r^2}{(t+1)^3}[\/latex]<\/li>\r\n \t
                  2. The volume is decreasing at a rate of [latex]-\\frac{\\pi}{36} \\, \\text{ft}^3\/\\text{min}[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                  3. [latex] \\approx 2.3[\/latex] ft\/hr<\/li>\r\n<\/ol>\r\n

                    Derivatives of Inverse Functions<\/h2>\r\n
                      \r\n \t
                    1. \r\n
                        \r\n \t
                      1. \"A<\/li>\r\n \t
                      2. [latex](f^{-1})^{\\prime}(1) \\approx 2[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                      3. \r\n
                          \r\n \t
                        1. \"A<\/li>\r\n \t
                        2. [latex](f^{-1})^{\\prime}(1) \\approx -1\/\\sqrt{3}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                        3. \r\n
                            \r\n \t
                          1. [latex]6[\/latex]<\/li>\r\n \t
                          2. [latex]x=f^{-1}(y)=(\\frac{y+3}{2})^{1\/3}[\/latex]<\/li>\r\n \t
                          3. [latex]\\frac{1}{6}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                          4. \r\n
                              \r\n \t
                            1. [latex]1[\/latex]<\/li>\r\n \t
                            2. [latex]x=f^{-1}(y)= \\sin^{-1} y[\/latex]<\/li>\r\n \t
                            3. [latex]1[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                            4. [latex]\\frac{1}{5}[\/latex]<\/li>\r\n \t
                            5. [latex]\\frac{1}{3}[\/latex]<\/li>\r\n \t
                            6. [latex]1[\/latex]<\/li>\r\n \t
                            7. \r\n
                                \r\n \t
                              1. [latex]4[\/latex]<\/li>\r\n \t
                              2. [latex]y=4x[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                              3. \r\n
                                  \r\n \t
                                1. [latex]-\\frac{1}{13}[\/latex]<\/li>\r\n \t
                                2. [latex]y=-\\frac{1}{13}x+\\frac{18}{13}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                                3. [latex]\\large \\frac{2x}{\\sqrt{1-x^4}}[\/latex]<\/li>\r\n \t
                                4. [latex]\\large \\frac{-1}{\\sqrt{1-x^2}}[\/latex]<\/li>\r\n \t
                                5. [latex]\\large \\frac{3(1 + \\tan^{-1} x)^2}{1+x^2}[\/latex]<\/li>\r\n \t
                                6. [latex]\\large \\frac{-1}{(1+x^2)(\\tan^{-1} x)^2}[\/latex]<\/li>\r\n \t
                                7. [latex]\\large \\frac{x}{(5-x^2)\\sqrt{4-x^2}}[\/latex]<\/li>\r\n \t
                                8. -[latex]1[\/latex]<\/li>\r\n \t
                                9. [latex]\\frac{1}{2}[\/latex]<\/li>\r\n \t
                                10. [latex]\\frac{1}{10}[\/latex]<\/li>\r\n \t
                                11. \r\n
                                    \r\n \t
                                  1. [latex]v(t)=\\frac{1}{1+t^2}[\/latex]<\/li>\r\n \t
                                  2. [latex]a(t)=\\frac{-2t}{(1+t^2)^2}[\/latex]<\/li>\r\n \t
                                  3. [latex]v(2)=0.2, \\, v(4)=0.06, \\, v(6)=0.03; \\, a(2)=-0.16, \\, a(4)=-0.028, \\, a(6)=-0.0088[\/latex]<\/li>\r\n \t
                                  4. The hockey puck is decelerating\/slowing down at [latex]2[\/latex], [latex]4[\/latex], and [latex]6[\/latex] seconds.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                                  5. -[latex]0.0168[\/latex] radians per foot<\/li>\r\n \t
                                  6. \r\n
                                      \r\n \t
                                    1. [latex]\\frac{d\\theta}{dx}=\\frac{10}{100+x^2}-\\frac{40}{1600+x^2}[\/latex]<\/li>\r\n \t
                                    2. [latex]\\frac{18}{325}, \\, \\frac{9}{340}, \\, \\frac{42}{4745}, \\, 0[\/latex]<\/li>\r\n \t
                                    3. As a person moves farther away from the screen, the viewing angle is increasing, which implies that as he or she moves farther away, his or her screen vision is widening.<\/li>\r\n \t
                                    4. [latex]-\\frac{54}{12905}, \\, -\\frac{3}{500}, \\, -\\frac{198}{29945}, \\, -\\frac{9}{1360}[\/latex]<\/li>\r\n \t
                                    5. As the person moves beyond [latex]20[\/latex] feet from the screen, the viewing angle is decreasing. The optimal distance the person should sit for maximizing the viewing angle is [latex]20[\/latex] feet<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n

                                      Implicit Differentiation<\/h2>\r\n
                                        \r\n \t
                                      1. [latex]\\frac{dy}{dx}=\\frac{-2x}{y}[\/latex]<\/li>\r\n \t
                                      2. [latex]\\frac{dy}{dx}=\\frac{x}{3y}-\\frac{y}{2x}[\/latex]<\/li>\r\n \t
                                      3. [latex]\\frac{dy}{dx}=\\large \\frac{y-\\frac{y}{2\\sqrt{x+4}}}{\\sqrt{x+4}-x}[\/latex]<\/li>\r\n \t
                                      4. [latex]\\frac{dy}{dx}=\\large \\frac{y^2 \\cos(xy)}{2y- \\sin(xy)-xy \\cos xy}[\/latex]<\/li>\r\n \t
                                      5. [latex]\\frac{dy}{dx}=\\large \\frac{-3x^2 y-y^3}{x^3+3xy^2}[\/latex]<\/li>\r\n \t
                                      6. \"The [latex]y=-\\frac{1}{2}x+2[\/latex]<\/li>\r\n \t
                                      7. \"The [latex]y=\\large \\frac{1}{\\pi +12}x-\\frac{3\\pi +38}{\\pi +12}[\/latex]<\/li>\r\n \t
                                      8. \"The [latex]y=0[\/latex]<\/li>\r\n \t
                                      9. \r\n
                                          \r\n \t
                                        1. [latex]y=\u2212x+2[\/latex]<\/li>\r\n \t
                                        2. [latex](3,-1)[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                                        3. \r\n
                                            \r\n \t
                                          1. [latex](\\pm \\sqrt{7},0)[\/latex]<\/li>\r\n \t
                                          2. Slope is [latex]-2[\/latex] at both intercepts<\/li>\r\n \t
                                          3. They are parallel since the slope is the same at both intercepts.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                                          4. [latex]y=\u2212x+1[\/latex]<\/li>\r\n \t
                                          5. \r\n
                                              \r\n \t
                                            1. -[latex]0.5926[\/latex]<\/li>\r\n \t
                                            2. When [latex]$81[\/latex] is spent on labor and [latex]$16[\/latex] is spent on capital, the amount spent on capital is decreasing by [latex]$0.5926[\/latex] per [latex]$1[\/latex] spent on labor.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                                            3. [latex]\\frac{dy}{dx}=-8[\/latex]<\/li>\r\n \t
                                            4. [latex]\\frac{dy}{dx}=-2.67[\/latex]<\/li>\r\n \t
                                            5. [latex]y^{\\prime}=-\\frac{1}{\\sqrt{1-x^2}}[\/latex]<\/li>\r\n<\/ol>\r\n

                                              Derivatives of Exponential and Logarithmic Functions<\/h2>\r\n
                                                \r\n \t
                                              1. [latex]f^{\\prime}(x) = 2xe^x+x^2 e^x[\/latex]<\/li>\r\n \t
                                              2. [latex]f^{\\prime}(x) = e^{x^3 \\ln x}(3x^2 \\ln x+x^2)[\/latex]<\/li>\r\n \t
                                              3. [latex]f^{\\prime}(x) = \\dfrac{4}{(e^x+e^{\u2212x})^2}[\/latex]<\/li>\r\n \t
                                              4. [latex]f^{\\prime}(x) = 2^{4x+2} \\cdot \\ln 2+8x[\/latex]<\/li>\r\n \t
                                              5. [latex]f^{\\prime}(x) = \\pi x^{\\pi -1} \\cdot \\pi^x + x^{\\pi} \\cdot \\pi^x \\ln \\pi [\/latex]<\/li>\r\n \t
                                              6. [latex]f^{\\prime}(x) = \\frac{5}{2(5x-7)}[\/latex]<\/li>\r\n \t
                                              7. [latex]f^{\\prime}(x) = \\frac{\\tan x}{\\ln 10}[\/latex]<\/li>\r\n \t
                                              8. [latex]f^{\\prime}(x) = 2^x \\cdot \\ln 2 \\cdot \\log_3 7^{x^2-4} + 2^x \\cdot \\frac{2x \\ln 7}{\\ln 3}[\/latex]<\/li>\r\n \t
                                              9. [latex]\\frac{dy}{dx} = (\\sin 2x)^{4x} [4 \\cdot \\ln(\\sin 2x) + 8x \\cdot \\cot 2x][\/latex]<\/li>\r\n \t
                                              10. [latex]\\frac{dy}{dx} = x^{\\log_2 x} \\cdot \\frac{2 \\ln x}{x \\ln 2}[\/latex]<\/li>\r\n \t
                                              11. [latex]\\frac{dy}{dx} = x^{\\cot x} \\cdot [\u2212\\csc^2 x \\cdot \\ln x+\\frac{\\cot x}{x}][\/latex]<\/li>\r\n \t
                                              12. [latex]\\frac{dy}{dx} = x^{-1\/2}(x^2+3)^{2\/3}(3x-4)^4 \\cdot [\\frac{-1}{2x}+\\frac{4x}{3(x^2+3)}+\\frac{12}{3x-4}][\/latex]<\/li>\r\n \t
                                              13. \"The [latex]y=\\frac{-1}{5+5 \\ln 5}x+(5+\\frac{1}{5+5 \\ln 5})[\/latex]<\/li>\r\n \t
                                              14. \r\n
                                                  \r\n \t
                                                1. [latex]x=e \\approx 2.718[\/latex]<\/li>\r\n \t
                                                2. [latex]y^{\\prime}>0[\/latex] on [latex](e,\\infty)[\/latex], and [latex]y^{\\prime}<0[\/latex] on [latex](0,e)[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                                                3. \r\n
                                                    \r\n \t
                                                  1. [latex]P=500,000(1.05)^t[\/latex] individuals<\/li>\r\n \t
                                                  2. [latex]P^{\\prime}(t)=24395 \\cdot (1.05)^t[\/latex] individuals per year<\/li>\r\n \t
                                                  3. [latex]39,737[\/latex] individuals per year<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                                                  4. \r\n
                                                      \r\n \t
                                                    1. At the beginning of 1960 there were [latex]5.3[\/latex] thousand cases of the disease in New York City. At the beginning of 1963 there were approximately [latex]723[\/latex] cases of the disease in New York City.<\/li>\r\n \t
                                                    2. At the beginning of 1960 the number of cases of the disease was decreasing at rate of [latex]-4.611[\/latex] thousand per year; at the beginning of 1963, the number of cases of the disease was decreasing at a rate of [latex]-0.2808[\/latex] thousand per year.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                                                    3. [latex]p=35741(1.045)^t[\/latex]<\/li>\r\n \t
                                                    4. <\/li>\r\n \t
                                                    5. \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                                                      Years since 1790<\/th>\r\n[latex]P''[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
                                                      [latex]0[\/latex]<\/td>\r\n[latex]69.25[\/latex]<\/td>\r\n<\/tr>\r\n
                                                      [latex]10[\/latex]<\/td>\r\n[latex]107.5[\/latex]<\/td>\r\n<\/tr>\r\n
                                                      [latex]20[\/latex]<\/td>\r\n[latex]167.0[\/latex]<\/td>\r\n<\/tr>\r\n
                                                      [latex]30[\/latex]<\/td>\r\n[latex]259.4[\/latex]<\/td>\r\n<\/tr>\r\n
                                                      [latex]40[\/latex]<\/td>\r\n[latex]402.8[\/latex]<\/td>\r\n<\/tr>\r\n
                                                      [latex]50[\/latex]<\/td>\r\n[latex]625.5[\/latex]<\/td>\r\n<\/tr>\r\n
                                                      [latex]60[\/latex]<\/td>\r\n[latex]971.4[\/latex]<\/td>\r\n<\/tr>\r\n
                                                      [latex]70[\/latex]<\/td>\r\n[latex]1508.5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t
                                                    6. <\/li>\r\n<\/ol>","rendered":"

                                                      Derivatives of Trigonometric Functions<\/h2>\n
                                                        \n
                                                      1. [latex]\\frac{dy}{dx}=2x- \\sec x \\tan x[\/latex]<\/li>\n
                                                      2. [latex]\\frac{dy}{dx}=2x \\cot x-x^2 \\csc^2 x[\/latex]<\/li>\n
                                                      3. [latex]\\frac{dy}{dx}=\\frac{x \\sec x \\tan x- \\sec x}{x^2}[\/latex]<\/li>\n
                                                      4. [latex]\\frac{dy}{dx}=(1- \\sin x)(1- \\sin x)- \\cos x(x+ \\cos x)[\/latex]<\/li>\n
                                                      5. [latex]\\frac{dy}{dx}=\\frac{2 \\csc^2 x}{(1+ \\cot x)^2}[\/latex]<\/li>\n
                                                      6. \n

                                                        [latex]y=\u2212x[\/latex]<\/p>\n

                                                        \"The<\/li>\n

                                                      7. \n

                                                        [latex]y=x+\\frac{2-3\\pi}{2}[\/latex]<\/p>\n

                                                        \"The<\/li>\n

                                                      8. [latex]y=\u2212x[\/latex]\n

                                                        \"The<\/li>\n

                                                      9. [latex]\\frac{d^2 y}{dx^2} = 3 \\cos x-x \\sin x[\/latex]<\/li>\n
                                                      10. [latex]\\frac{d^2 y}{dx^2} = \\frac{1}{2} \\sin x[\/latex]<\/li>\n
                                                      11. [latex]\\frac{d^2 y}{dx^2} = 2\\csc x( \\csc^2 x + \\cot^2 x)[\/latex]<\/li>\n
                                                      12. [latex]x = \\frac{(2n+1)\\pi}{4}[\/latex], where [latex]n[\/latex] is an integer<\/li>\n
                                                      13. [latex](\\frac{\\pi}{4},1), \\, (\\frac{3\\pi}{4},-1)[\/latex]<\/li>\n
                                                      14. [latex]a=0, \\, b=3[\/latex]<\/li>\n
                                                      15. [latex]y^{\\prime}=5 \\cos (x)[\/latex], increasing on [latex](0,\\frac{\\pi}{2}), \\, (\\frac{3\\pi}{2},\\frac{5\\pi}{2})[\/latex], and [latex](\\frac{7\\pi}{2},12)[\/latex]<\/li>\n
                                                      16. [latex]\\frac{d^3 y}{dx^3} = 3 \\sin x[\/latex]<\/li>\n
                                                      17. [latex]\\frac{d^4 y}{dx^4} = 5 \\cos x[\/latex]<\/li>\n
                                                      18. [latex]\\frac{d^3 y}{dx^3} = 720x^7-5 \\tan (x) \\sec^3 (x)- \\tan^3 (x) \\sec (x)[\/latex]<\/li>\n<\/ol>\n

                                                        The Chain Rule<\/h2>\n
                                                          \n
                                                        1. [latex]\\frac{dy}{dx} = 18u^2 \\cdot 7=18(7x-4)^2 \\cdot 7[\/latex]<\/li>\n
                                                        2. [latex]\\frac{dy}{dx} = \u2212\\sin u \\cdot \\frac{-1}{8}=\u2212\\sin (\\frac{\u2212x}{8}) \\cdot \\frac{-1}{8}[\/latex]<\/li>\n
                                                        3. [latex]\\frac{dy}{dx} = \\frac{8x-24}{2\\sqrt{4u+3}}=\\frac{4x-12}{\\sqrt{4x^2-24x+3}}[\/latex]<\/li>\n
                                                        4. \n
                                                            \n
                                                          1. [latex]f(u) = u^3, \\, u=3x^2+1[\/latex]<\/li>\n
                                                          2. [latex]\\frac{dy}{dx} = 18x(3x^2+1)^2[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                          3. \n
                                                              \n
                                                            1. [latex]f(u)=u^7, \\, u=\\frac{x}{7}+\\frac{7}{x}[\/latex]<\/li>\n
                                                            2. [latex]\\frac{dy}{dx} = 7(\\frac{x}{7}+\\frac{7}{x})^6 \\cdot (\\frac{1}{7}-\\frac{7}{x^2})[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                            3. \n
                                                                \n
                                                              1. [latex]f(u)= \\csc u, \\, u=\\pi x+1[\/latex]<\/li>\n
                                                              2. [latex]\\frac{dy}{dx} = \u2212\\pi \\csc (\\pi x+1) \\cdot \\cot (\\pi x+1)[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                              3. \n
                                                                  \n
                                                                1. [latex]f(u)=-6u^{-3}, \\, u= \\sin x[\/latex]<\/li>\n
                                                                2. [latex]\\frac{dy}{dx} = 18 \\sin^{-4} x \\cdot \\cos x[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                                3. [latex]\\frac{dy}{dx} = \\frac{4}{(5-2x)^3}[\/latex]<\/li>\n
                                                                4. [latex]\\frac{dy}{dx} = 6(2x^3-x^2+6x+1)^2(3x^2-x+3)[\/latex]<\/li>\n
                                                                5. [latex]\\frac{dy}{dx} = -3(\\tan x+ \\sin x)^{-4} \\cdot (\\sec^2 x+ \\cos x)[\/latex]<\/li>\n
                                                                6. [latex]\\frac{dy}{dx} = -7 \\cos (\\cos 7x) \\cdot \\sin 7x[\/latex]<\/li>\n
                                                                7. [latex]\\frac{dy}{dx} = -12 \\cot^2 (4x+1) \\cdot \\csc^2 (4x+1)[\/latex]<\/li>\n
                                                                8. [latex]10[\/latex]<\/li>\n
                                                                9. [latex]-\\frac{1}{8}[\/latex]<\/li>\n
                                                                10. –[latex]4[\/latex]<\/li>\n
                                                                11. –[latex]12[\/latex]<\/li>\n
                                                                12. [latex]10\\frac{3}{4}[\/latex]<\/li>\n
                                                                13. [latex]y=-\\frac{1}{2}x[\/latex]<\/li>\n
                                                                14. [latex]x= \\pm \\sqrt{6}[\/latex]<\/li>\n
                                                                15. \n
                                                                    \n
                                                                  1. [latex]-\\frac{200}{343}[\/latex] m\/s<\/li>\n
                                                                  2. [latex]\\frac{600}{2401} \\, \\text{m\/s}^2[\/latex]<\/li>\n
                                                                  3. The train is slowing down since velocity and acceleration have opposite signs<\/li>\n<\/ol>\n<\/li>\n
                                                                  4. \n
                                                                      \n
                                                                    1. [latex]C^{\\prime}(x)=0.0003x^2-0.04x+3[\/latex]<\/li>\n
                                                                    2. [latex]\\frac{dC}{dt}=100 \\cdot (0.0003x^2-0.04x+3)[\/latex]<\/li>\n
                                                                    3. Approximately [latex]$90,300[\/latex] per week<\/li>\n<\/ol>\n<\/li>\n
                                                                    4. \n
                                                                        \n
                                                                      1. [latex]\\frac{dS}{dt}=-\\frac{8\\pi r^2}{(t+1)^3}[\/latex]<\/li>\n
                                                                      2. The volume is decreasing at a rate of [latex]-\\frac{\\pi}{36} \\, \\text{ft}^3\/\\text{min}[\/latex].<\/li>\n<\/ol>\n<\/li>\n
                                                                      3. [latex]\\approx 2.3[\/latex] ft\/hr<\/li>\n<\/ol>\n

                                                                        Derivatives of Inverse Functions<\/h2>\n
                                                                          \n
                                                                        1. \n
                                                                            \n
                                                                          1. \"A<\/li>\n
                                                                          2. [latex](f^{-1})^{\\prime}(1) \\approx 2[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                                          3. \n
                                                                              \n
                                                                            1. \"A<\/li>\n
                                                                            2. [latex](f^{-1})^{\\prime}(1) \\approx -1\/\\sqrt{3}[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                                            3. \n
                                                                                \n
                                                                              1. [latex]6[\/latex]<\/li>\n
                                                                              2. [latex]x=f^{-1}(y)=(\\frac{y+3}{2})^{1\/3}[\/latex]<\/li>\n
                                                                              3. [latex]\\frac{1}{6}[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                                              4. \n
                                                                                  \n
                                                                                1. [latex]1[\/latex]<\/li>\n
                                                                                2. [latex]x=f^{-1}(y)= \\sin^{-1} y[\/latex]<\/li>\n
                                                                                3. [latex]1[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                                                4. [latex]\\frac{1}{5}[\/latex]<\/li>\n
                                                                                5. [latex]\\frac{1}{3}[\/latex]<\/li>\n
                                                                                6. [latex]1[\/latex]<\/li>\n
                                                                                7. \n
                                                                                    \n
                                                                                  1. [latex]4[\/latex]<\/li>\n
                                                                                  2. [latex]y=4x[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                                                  3. \n
                                                                                      \n
                                                                                    1. [latex]-\\frac{1}{13}[\/latex]<\/li>\n
                                                                                    2. [latex]y=-\\frac{1}{13}x+\\frac{18}{13}[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                                                    3. [latex]\\large \\frac{2x}{\\sqrt{1-x^4}}[\/latex]<\/li>\n
                                                                                    4. [latex]\\large \\frac{-1}{\\sqrt{1-x^2}}[\/latex]<\/li>\n
                                                                                    5. [latex]\\large \\frac{3(1 + \\tan^{-1} x)^2}{1+x^2}[\/latex]<\/li>\n
                                                                                    6. [latex]\\large \\frac{-1}{(1+x^2)(\\tan^{-1} x)^2}[\/latex]<\/li>\n
                                                                                    7. [latex]\\large \\frac{x}{(5-x^2)\\sqrt{4-x^2}}[\/latex]<\/li>\n
                                                                                    8. –[latex]1[\/latex]<\/li>\n
                                                                                    9. [latex]\\frac{1}{2}[\/latex]<\/li>\n
                                                                                    10. [latex]\\frac{1}{10}[\/latex]<\/li>\n
                                                                                    11. \n
                                                                                        \n
                                                                                      1. [latex]v(t)=\\frac{1}{1+t^2}[\/latex]<\/li>\n
                                                                                      2. [latex]a(t)=\\frac{-2t}{(1+t^2)^2}[\/latex]<\/li>\n
                                                                                      3. [latex]v(2)=0.2, \\, v(4)=0.06, \\, v(6)=0.03; \\, a(2)=-0.16, \\, a(4)=-0.028, \\, a(6)=-0.0088[\/latex]<\/li>\n
                                                                                      4. The hockey puck is decelerating\/slowing down at [latex]2[\/latex], [latex]4[\/latex], and [latex]6[\/latex] seconds.<\/li>\n<\/ol>\n<\/li>\n
                                                                                      5. –[latex]0.0168[\/latex] radians per foot<\/li>\n
                                                                                      6. \n
                                                                                          \n
                                                                                        1. [latex]\\frac{d\\theta}{dx}=\\frac{10}{100+x^2}-\\frac{40}{1600+x^2}[\/latex]<\/li>\n
                                                                                        2. [latex]\\frac{18}{325}, \\, \\frac{9}{340}, \\, \\frac{42}{4745}, \\, 0[\/latex]<\/li>\n
                                                                                        3. As a person moves farther away from the screen, the viewing angle is increasing, which implies that as he or she moves farther away, his or her screen vision is widening.<\/li>\n
                                                                                        4. [latex]-\\frac{54}{12905}, \\, -\\frac{3}{500}, \\, -\\frac{198}{29945}, \\, -\\frac{9}{1360}[\/latex]<\/li>\n
                                                                                        5. As the person moves beyond [latex]20[\/latex] feet from the screen, the viewing angle is decreasing. The optimal distance the person should sit for maximizing the viewing angle is [latex]20[\/latex] feet<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

                                                                                          Implicit Differentiation<\/h2>\n
                                                                                            \n
                                                                                          1. [latex]\\frac{dy}{dx}=\\frac{-2x}{y}[\/latex]<\/li>\n
                                                                                          2. [latex]\\frac{dy}{dx}=\\frac{x}{3y}-\\frac{y}{2x}[\/latex]<\/li>\n
                                                                                          3. [latex]\\frac{dy}{dx}=\\large \\frac{y-\\frac{y}{2\\sqrt{x+4}}}{\\sqrt{x+4}-x}[\/latex]<\/li>\n
                                                                                          4. [latex]\\frac{dy}{dx}=\\large \\frac{y^2 \\cos(xy)}{2y- \\sin(xy)-xy \\cos xy}[\/latex]<\/li>\n
                                                                                          5. [latex]\\frac{dy}{dx}=\\large \\frac{-3x^2 y-y^3}{x^3+3xy^2}[\/latex]<\/li>\n
                                                                                          6. \"The [latex]y=-\\frac{1}{2}x+2[\/latex]<\/li>\n
                                                                                          7. \"The [latex]y=\\large \\frac{1}{\\pi +12}x-\\frac{3\\pi +38}{\\pi +12}[\/latex]<\/li>\n
                                                                                          8. \"The [latex]y=0[\/latex]<\/li>\n
                                                                                          9. \n
                                                                                              \n
                                                                                            1. [latex]y=\u2212x+2[\/latex]<\/li>\n
                                                                                            2. [latex](3,-1)[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                                                            3. \n
                                                                                                \n
                                                                                              1. [latex](\\pm \\sqrt{7},0)[\/latex]<\/li>\n
                                                                                              2. Slope is [latex]-2[\/latex] at both intercepts<\/li>\n
                                                                                              3. They are parallel since the slope is the same at both intercepts.<\/li>\n<\/ol>\n<\/li>\n
                                                                                              4. [latex]y=\u2212x+1[\/latex]<\/li>\n
                                                                                              5. \n
                                                                                                  \n
                                                                                                1. –[latex]0.5926[\/latex]<\/li>\n
                                                                                                2. When [latex]$81[\/latex] is spent on labor and [latex]$16[\/latex] is spent on capital, the amount spent on capital is decreasing by [latex]$0.5926[\/latex] per [latex]$1[\/latex] spent on labor.<\/li>\n<\/ol>\n<\/li>\n
                                                                                                3. [latex]\\frac{dy}{dx}=-8[\/latex]<\/li>\n
                                                                                                4. [latex]\\frac{dy}{dx}=-2.67[\/latex]<\/li>\n
                                                                                                5. [latex]y^{\\prime}=-\\frac{1}{\\sqrt{1-x^2}}[\/latex]<\/li>\n<\/ol>\n

                                                                                                  Derivatives of Exponential and Logarithmic Functions<\/h2>\n
                                                                                                    \n
                                                                                                  1. [latex]f^{\\prime}(x) = 2xe^x+x^2 e^x[\/latex]<\/li>\n
                                                                                                  2. [latex]f^{\\prime}(x) = e^{x^3 \\ln x}(3x^2 \\ln x+x^2)[\/latex]<\/li>\n
                                                                                                  3. [latex]f^{\\prime}(x) = \\dfrac{4}{(e^x+e^{\u2212x})^2}[\/latex]<\/li>\n
                                                                                                  4. [latex]f^{\\prime}(x) = 2^{4x+2} \\cdot \\ln 2+8x[\/latex]<\/li>\n
                                                                                                  5. [latex]f^{\\prime}(x) = \\pi x^{\\pi -1} \\cdot \\pi^x + x^{\\pi} \\cdot \\pi^x \\ln \\pi[\/latex]<\/li>\n
                                                                                                  6. [latex]f^{\\prime}(x) = \\frac{5}{2(5x-7)}[\/latex]<\/li>\n
                                                                                                  7. [latex]f^{\\prime}(x) = \\frac{\\tan x}{\\ln 10}[\/latex]<\/li>\n
                                                                                                  8. [latex]f^{\\prime}(x) = 2^x \\cdot \\ln 2 \\cdot \\log_3 7^{x^2-4} + 2^x \\cdot \\frac{2x \\ln 7}{\\ln 3}[\/latex]<\/li>\n
                                                                                                  9. [latex]\\frac{dy}{dx} = (\\sin 2x)^{4x} [4 \\cdot \\ln(\\sin 2x) + 8x \\cdot \\cot 2x][\/latex]<\/li>\n
                                                                                                  10. [latex]\\frac{dy}{dx} = x^{\\log_2 x} \\cdot \\frac{2 \\ln x}{x \\ln 2}[\/latex]<\/li>\n
                                                                                                  11. [latex]\\frac{dy}{dx} = x^{\\cot x} \\cdot [\u2212\\csc^2 x \\cdot \\ln x+\\frac{\\cot x}{x}][\/latex]<\/li>\n
                                                                                                  12. [latex]\\frac{dy}{dx} = x^{-1\/2}(x^2+3)^{2\/3}(3x-4)^4 \\cdot [\\frac{-1}{2x}+\\frac{4x}{3(x^2+3)}+\\frac{12}{3x-4}][\/latex]<\/li>\n
                                                                                                  13. \"The [latex]y=\\frac{-1}{5+5 \\ln 5}x+(5+\\frac{1}{5+5 \\ln 5})[\/latex]<\/li>\n
                                                                                                  14. \n
                                                                                                      \n
                                                                                                    1. [latex]x=e \\approx 2.718[\/latex]<\/li>\n
                                                                                                    2. [latex]y^{\\prime}>0[\/latex] on [latex](e,\\infty)[\/latex], and [latex]y^{\\prime}<0[\/latex] on [latex](0,e)[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                                                                                                    3. \n
                                                                                                        \n
                                                                                                      1. [latex]P=500,000(1.05)^t[\/latex] individuals<\/li>\n
                                                                                                      2. [latex]P^{\\prime}(t)=24395 \\cdot (1.05)^t[\/latex] individuals per year<\/li>\n
                                                                                                      3. [latex]39,737[\/latex] individuals per year<\/li>\n<\/ol>\n<\/li>\n
                                                                                                      4. \n
                                                                                                          \n
                                                                                                        1. At the beginning of 1960 there were [latex]5.3[\/latex] thousand cases of the disease in New York City. At the beginning of 1963 there were approximately [latex]723[\/latex] cases of the disease in New York City.<\/li>\n
                                                                                                        2. At the beginning of 1960 the number of cases of the disease was decreasing at rate of [latex]-4.611[\/latex] thousand per year; at the beginning of 1963, the number of cases of the disease was decreasing at a rate of [latex]-0.2808[\/latex] thousand per year.<\/li>\n<\/ol>\n<\/li>\n
                                                                                                        3. [latex]p=35741(1.045)^t[\/latex]<\/li>\n
                                                                                                        4. <\/li>\n
                                                                                                        5. \n\n\n\n\n\n\n\n\n\n\n\n\n
                                                                                                          Years since 1790<\/th>\n[latex]P''[\/latex]<\/th>\n<\/tr>\n<\/thead>\n
                                                                                                          [latex]0[\/latex]<\/td>\n[latex]69.25[\/latex]<\/td>\n<\/tr>\n
                                                                                                          [latex]10[\/latex]<\/td>\n[latex]107.5[\/latex]<\/td>\n<\/tr>\n
                                                                                                          [latex]20[\/latex]<\/td>\n[latex]167.0[\/latex]<\/td>\n<\/tr>\n
                                                                                                          [latex]30[\/latex]<\/td>\n[latex]259.4[\/latex]<\/td>\n<\/tr>\n
                                                                                                          [latex]40[\/latex]<\/td>\n[latex]402.8[\/latex]<\/td>\n<\/tr>\n
                                                                                                          [latex]50[\/latex]<\/td>\n[latex]625.5[\/latex]<\/td>\n<\/tr>\n
                                                                                                          [latex]60[\/latex]<\/td>\n[latex]971.4[\/latex]<\/td>\n<\/tr>\n
                                                                                                          [latex]70[\/latex]<\/td>\n[latex]1508.5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n
                                                                                                        6. <\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":6,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":108,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/153"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/153\/revisions"}],"predecessor-version":[{"id":166,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/153\/revisions\/166"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/108"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/153\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=153"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=153"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=153"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=153"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}