Techniques for Differentiation: Get Stronger Answer Key

Derivatives of Trigonometric Functions

  1. [latex]\frac{dy}{dx}=2x- \sec x \tan x[/latex]
  2. [latex]\frac{dy}{dx}=2x \cot x-x^2 \csc^2 x[/latex]
  3. [latex]\frac{dy}{dx}=\frac{x \sec x \tan x- \sec x}{x^2}[/latex]
  4. [latex]\frac{dy}{dx}=(1- \sin x)(1- \sin x)- \cos x(x+ \cos x)[/latex]
  5. [latex]\frac{dy}{dx}=\frac{2 \csc^2 x}{(1+ \cot x)^2}[/latex]

  6. [latex]y=−x[/latex]

    The graph shows negative sin(x) and the straight line T(x) with slope −1 and y intercept 0.

  7. [latex]y=x+\frac{2-3\pi}{2}[/latex]

    The graph shows the cosine function shifted up one and has the straight line T(x) with slope 1 and y intercept (2 – 3π)/2.

  8. [latex]y=−x[/latex]

    The graph shows the function as starting at (−1, 3), decreasing to the origin, continuing to slowly decrease to about (1, −0.5), at which point it decreases very quickly.

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

The Chain Rule

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

Derivatives of Inverse Functions

    1. A curved line starting at (−3, 0) and passing through (−2, 1) and (1, 2). There is another curved line that is symmetric with this about the line x = y. That is, it starts at (0, −3) and passes through (1, −2) and (2, 1).
    2. [latex](f^{-1})^{\prime}(1) \approx 2[/latex]
    1. A quarter circle starting at (0, 4) and ending at (4, 0).
    2. [latex](f^{-1})^{\prime}(1) \approx -1/\sqrt{3}[/latex]
    1. [latex]6[/latex]
    2. [latex]x=f^{-1}(y)=(\frac{y+3}{2})^{1/3}[/latex]
    3. [latex]\frac{1}{6}[/latex]
    1. [latex]1[/latex]
    2. [latex]x=f^{-1}(y)= \sin^{-1} y[/latex]
    3. [latex]1[/latex]
  5. [latex]\frac{1}{5}[/latex]
  6. [latex]\frac{1}{3}[/latex]
  7. [latex]1[/latex]
    1. [latex]4[/latex]
    2. [latex]y=4x[/latex]
    1. [latex]-\frac{1}{13}[/latex]
    2. [latex]y=-\frac{1}{13}x+\frac{18}{13}[/latex]
  10. [latex]\large \frac{2x}{\sqrt{1-x^4}}[/latex]
  11. [latex]\large \frac{-1}{\sqrt{1-x^2}}[/latex]
  12. [latex]\large \frac{3(1 + \tan^{-1} x)^2}{1+x^2}[/latex]
  13. [latex]\large \frac{-1}{(1+x^2)(\tan^{-1} x)^2}[/latex]
  14. [latex]\large \frac{x}{(5-x^2)\sqrt{4-x^2}}[/latex]
  15. –[latex]1[/latex]
  16. [latex]\frac{1}{2}[/latex]
  17. [latex]\frac{1}{10}[/latex]
    1. [latex]v(t)=\frac{1}{1+t^2}[/latex]
    2. [latex]a(t)=\frac{-2t}{(1+t^2)^2}[/latex]
    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]
    4. The hockey puck is decelerating/slowing down at [latex]2[/latex], [latex]4[/latex], and [latex]6[/latex] seconds.
  19. –[latex]0.0168[/latex] radians per foot
    1. [latex]\frac{d\theta}{dx}=\frac{10}{100+x^2}-\frac{40}{1600+x^2}[/latex]
    2. [latex]\frac{18}{325}, \, \frac{9}{340}, \, \frac{42}{4745}, \, 0[/latex]
    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.
    4. [latex]-\frac{54}{12905}, \, -\frac{3}{500}, \, -\frac{198}{29945}, \, -\frac{9}{1360}[/latex]
    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

Implicit Differentiation

  1. [latex]\frac{dy}{dx}=\frac{-2x}{y}[/latex]
  2. [latex]\frac{dy}{dx}=\frac{x}{3y}-\frac{y}{2x}[/latex]
  3. [latex]\frac{dy}{dx}=\large \frac{y-\frac{y}{2\sqrt{x+4}}}{\sqrt{x+4}-x}[/latex]
  4. [latex]\frac{dy}{dx}=\large \frac{y^2 \cos(xy)}{2y- \sin(xy)-xy \cos xy}[/latex]
  5. [latex]\frac{dy}{dx}=\large \frac{-3x^2 y-y^3}{x^3+3xy^2}[/latex]
  6. The graph has a crescent in each of the four quadrants. There is a straight line marked T(x) with slope −1/2 and y intercept 2. [latex]y=-\frac{1}{2}x+2[/latex]
  7. The graph has two curves, one in the first quadrant and one in the fourth quadrant. They are symmetric about the x axis. The curve in the first quadrant goes from (0.3, 5) to (1.5, 3.5) to (5, 4). There is a straight line marked T(x) with slope 1/(π + 12) and y intercept −(3π + 38)/(π + 12). [latex]y=\large \frac{1}{\pi +12}x-\frac{3\pi +38}{\pi +12}[/latex]
  8. The graph starts in the third quadrant near (−5, 0), remains near 0 until x = −4, at which point it decreases until it reaches near (0, −5). There is an asymptote at x = 0. The graph begins again near (0, 5) decreases to (1, 0) and then increases a little bit before decreasing to be near (5, 0). There is a straight line marked T(x) that coincides with y = 0. [latex]y=0[/latex]
    1. [latex]y=−x+2[/latex]
    2. [latex](3,-1)[/latex]
    1. [latex](\pm \sqrt{7},0)[/latex]
    2. Slope is [latex]-2[/latex] at both intercepts
    3. They are parallel since the slope is the same at both intercepts.
  11. [latex]y=−x+1[/latex]
    1. –[latex]0.5926[/latex]
    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.
  13. [latex]\frac{dy}{dx}=-8[/latex]
  14. [latex]\frac{dy}{dx}=-2.67[/latex]
  15. [latex]y^{\prime}=-\frac{1}{\sqrt{1-x^2}}[/latex]

Derivatives of Exponential and Logarithmic Functions

  1. [latex]f^{\prime}(x) = 2xe^x+x^2 e^x[/latex]
  2. [latex]f^{\prime}(x) = e^{x^3 \ln x}(3x^2 \ln x+x^2)[/latex]
  3. [latex]f^{\prime}(x) = \dfrac{4}{(e^x+e^{−x})^2}[/latex]
  4. [latex]f^{\prime}(x) = 2^{4x+2} \cdot \ln 2+8x[/latex]
  5. [latex]f^{\prime}(x) = \pi x^{\pi -1} \cdot \pi^x + x^{\pi} \cdot \pi^x \ln \pi[/latex]
  6. [latex]f^{\prime}(x) = \frac{5}{2(5x-7)}[/latex]
  7. [latex]f^{\prime}(x) = \frac{\tan x}{\ln 10}[/latex]
  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]
  9. [latex]\frac{dy}{dx} = (\sin 2x)^{4x} [4 \cdot \ln(\sin 2x) + 8x \cdot \cot 2x][/latex]
  10. [latex]\frac{dy}{dx} = x^{\log_2 x} \cdot \frac{2 \ln x}{x \ln 2}[/latex]
  11. [latex]\frac{dy}{dx} = x^{\cot x} \cdot [−\csc^2 x \cdot \ln x+\frac{\cot x}{x}][/latex]
  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]
  13. The function starts at (−3, 0), decreases slightly and then increases through the origin and increases to (1.25, 10). There is a straight line marked T(x) with slope −1/(5 + 5 ln 5) and y intercept 5 + 1/(5 + 5 ln 5). [latex]y=\frac{-1}{5+5 \ln 5}x+(5+\frac{1}{5+5 \ln 5})[/latex]
    1. [latex]x=e \approx 2.718[/latex]
    2. [latex]y^{\prime}>0[/latex] on [latex](e,\infty)[/latex], and [latex]y^{\prime}<0[/latex] on [latex](0,e)[/latex]
    1. [latex]P=500,000(1.05)^t[/latex] individuals
    2. [latex]P^{\prime}(t)=24395 \cdot (1.05)^t[/latex] individuals per year
    3. [latex]39,737[/latex] individuals per year
    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.
    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.
  17. [latex]p=35741(1.045)^t[/latex]
  18.  
  19. Years since 1790 [latex]P''[/latex]
    [latex]0[/latex] [latex]69.25[/latex]
    [latex]10[/latex] [latex]107.5[/latex]
    [latex]20[/latex] [latex]167.0[/latex]
    [latex]30[/latex] [latex]259.4[/latex]
    [latex]40[/latex] [latex]402.8[/latex]
    [latex]50[/latex] [latex]625.5[/latex]
    [latex]60[/latex] [latex]971.4[/latex]
    [latex]70[/latex] [latex]1508.5[/latex]
  20.