{"id":3293,"date":"2024-06-18T18:06:25","date_gmt":"2024-06-18T18:06:25","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&p=3293"},"modified":"2024-08-05T12:45:26","modified_gmt":"2024-08-05T12:45:26","slug":"derivatives-of-trigonometric-functions-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/derivatives-of-trigonometric-functions-learn-it-2\/","title":{"raw":"Derivatives of Trigonometric Functions: Learn It 2","rendered":"Derivatives of Trigonometric Functions: Learn It 2"},"content":{"raw":"

Derivatives of Other Trigonometric Functions<\/h2>\r\n

To further explore the derivatives of trigonometric functions, we use the quotient rule and other calculus techniques since the remaining trigonometric functions are expressed as quotients involving sine and cosine.<\/p>\r\n

\r\n

Find the derivative of [latex]f(x)= \\tan x[\/latex].<\/p>\r\n

[reveal-answer q=\"fs-id1169739242784\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"fs-id1169739242784\"]<\/p>\r\n

Start by expressing [latex]\\tan x[\/latex] as the quotient of [latex]\\sin x[\/latex] and [latex]\\cos x[\/latex]:<\/p>\r\n

[latex]f(x)= \\tan x=\\dfrac{\\sin x}{\\cos x}[\/latex]<\/div>\r\n

Now apply the quotient rule to obtain<\/p>\r\n

[latex]f^{\\prime}(x)=\\dfrac{\\cos x \\cos x-(\u2212\\sin x)\\sin x}{(\\cos x)^2}[\/latex]<\/div>\r\n

Simplifying, we obtain<\/p>\r\n

[latex]f^{\\prime}(x)=\\dfrac{\\cos^2 x+\\sin^2 x}{\\cos^2 x}[\/latex]<\/div>\r\n

Recognizing that [latex]\\cos^2 x+\\sin^2 x=1[\/latex], by the Pythagorean Identity, we now have<\/p>\r\n

[latex]f^{\\prime}(x)=\\dfrac{1}{\\cos^2 x}[\/latex]<\/div>\r\n

Finally, use the identity [latex]\\sec x=\\dfrac{1}{\\cos x}[\/latex] to obtain<\/p>\r\n

[latex]f^{\\prime}(x)=\\sec^2 x[\/latex]<\/div>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n
\r\n

Find the derivative of [latex]f(x)= \\cot x[\/latex].<\/p>\r\n

[reveal-answer q=\"487336\"]Hint[\/reveal-answer]
\r\n[hidden-answer a=\"487336\"]<\/p>\r\n

Rewrite [latex]\\cot x[\/latex] as [latex]\\frac{\\cos x}{\\sin x}[\/latex] and use the quotient rule.<\/p>\r\n

[\/hidden-answer]<\/p>\r\n

[reveal-answer q=\"fs-id1169739301461\"]Show Solution[\/reveal-answer]
\r\n[hidden-answer a=\"fs-id1169739301461\"]<\/p>\r\n

[latex]f^{\\prime}(x)=\u2212\\csc^2 x[\/latex]<\/p>\r\n

[\/hidden-answer]<\/p>\r\n<\/section>\r\n

The derivatives of the remaining trigonometric functions may be obtained by using similar techniques.\u00a0<\/p>\r\n

\r\n

derivatives of\u00a0 [latex]\\tan x, \\, \\cot x, \\, \\sec x[\/latex], and [latex]\\csc x[\/latex]<\/strong><\/h3>\r\n
    \r\n\t
  • Derivative of Tangent:<\/strong>\r\n
    [latex]\\frac{d}{dx}(\\tan x)=\\sec^2 x[\/latex]<\/div>\r\n<\/li>\r\n\t
  • Derivative of Cotangent:<\/strong>\r\n
    [latex]\\frac{d}{dx}(\\cot x)=\u2212\\csc^2 x[\/latex]<\/div>\r\n<\/li>\r\n\t
  • Derivative of Secant:<\/strong>\r\n
    [latex]\\frac{d}{dx}(\\sec x)= \\sec x \\tan x[\/latex]<\/div>\r\n<\/li>\r\n\t
  • Derivative of Cosecant:<\/strong>\r\n
    [latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex]<\/div>\r\n<\/li>\r\n<\/ul>\r\n<\/section>\r\n
    \r\n

    As you navigate problems involving derivatives of trigonometric functions, don't forget our handy table of trigonometric function values of common angles:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    Angle<\/strong><\/td>\r\n [latex]0[\/latex] <\/strong><\/td>\r\n [latex]\\frac{\\pi }{6},\\text{ or }{30}^{\\circ}[\/latex] <\/strong><\/td>\r\n [latex]\\frac{\\pi }{4},\\text{ or } {45}^{\\circ }[\/latex] <\/strong><\/td>\r\n [latex]\\frac{\\pi }{3},\\text{ or }{60}^{\\circ }[\/latex] <\/strong><\/td>\r\n [latex]\\frac{\\pi }{2},\\text{ or }{90}^{\\circ }[\/latex] <\/strong><\/td>\r\n<\/tr>\r\n
    Cosine<\/strong><\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\r\n[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\r\n[latex]\\frac{1}{2}[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n
    Sine<\/strong><\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]\\frac{1}{2}[\/latex]<\/td>\r\n[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\r\n[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n
    Tangent<\/strong><\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]\\sqrt{3}[\/latex]<\/td>\r\nUndefined<\/td>\r\n<\/tr>\r\n
    Secant<\/strong><\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/td>\r\n[latex]\\sqrt{2}[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\nUndefined<\/td>\r\n<\/tr>\r\n
    Cosecant<\/strong><\/td>\r\nUndefined<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]\\sqrt{2}[\/latex]<\/td>\r\n[latex]\\frac{2\\sqrt{3}}{3}[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n
    Cotangent<\/strong><\/td>\r\nUndefined<\/td>\r\n[latex]\\sqrt{3}[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]\\frac{\\sqrt{3}}{3}[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>\r\n
    \r\n

    Find the equation of a line tangent to the graph of [latex]f(x)= \\cot x[\/latex] at [latex]x=\\dfrac{\\pi}{4}[\/latex].<\/p>\r\n

    [reveal-answer q=\"fs-id1169736658874\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"fs-id1169736658874\"]<\/p>\r\n

    To find the equation of the tangent line, we need a point and a slope at that point. To find the point, compute<\/p>\r\n

    [latex]f\\left(\\frac{\\pi}{4}\\right)= \\cot \\frac{\\pi}{4}=1[\/latex].<\/div>\r\n

    Thus the tangent line passes through the point [latex]\\left(\\frac{\\pi}{4},1\\right)[\/latex]. Next, find the slope by finding the derivative of [latex]f(x)= \\cot x[\/latex] and evaluating it at [latex]\\frac{\\pi}{4}[\/latex]:<\/p>\r\n

    [latex]f^{\\prime}(x)=\u2212\\csc^2 x[\/latex]\u00a0 and\u00a0 [latex]f^{\\prime}\\left(\\frac{\\pi}{4}\\right)=\u2212\\csc^2 \\left(\\frac{\\pi}{4}\\right)=-2[\/latex].<\/div>\r\n

    Using the point-slope equation of the line, we obtain<\/p>\r\n

    [latex]y-1=-2\\left(x-\\frac{\\pi}{4}\\right)[\/latex]<\/div>\r\n

    or equivalently,<\/p>\r\n

    [latex]y=-2x+1+\\frac{\\pi}{2}[\/latex].<\/div>\r\n\r\n\r\n[\/hidden-answer]<\/section>\r\n
    \r\n

    Find the derivative of [latex]f(x)= \\csc x+x \\tan x.[\/latex]<\/p>\r\n

    [reveal-answer q=\"fs-id1169736655897\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"fs-id1169736655897\"]<\/p>\r\n

    To find this derivative, we must use both the sum rule and the product rule. Using the sum rule, we find<\/p>\r\n

    [latex]f^{\\prime}(x)=\\frac{d}{dx}(\\csc x)+\\frac{d}{dx}(x \\tan x)[\/latex].<\/div>\r\n

     <\/p>\r\n

    In the first term, [latex]\\frac{d}{dx}(\\csc x)=\u2212\\csc x \\cot x[\/latex], and by applying the product rule to the second term we obtain<\/p>\r\n

    [latex]\\frac{d}{dx}(x \\tan x)=(1)(\\tan x)+(\\sec^2 x)(x)[\/latex].<\/div>\r\n

     <\/p>\r\n

    Therefore, we have<\/p>\r\n

    [latex]f^{\\prime}(x)=\u2212\\csc x \\cot x+ \\tan x+x \\sec^2 x[\/latex].<\/div>\r\n

    Watch the following video to see the worked solution to this example.<\/p>\r\n