Calculus Volume 1 · 3.5 Derivatives of Trigonometric Functions

Derivatives of Trigonometric Functions: practice problems

18 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

1. Exercises 175

   For the following exercises, find $\frac{dy}{dx}$ for the given functions.

   $y=x^2-\sec x+1$

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2. Exercises 177

   For the following exercises, find $\frac{dy}{dx}$ for the given functions.

   $y=x^2\cot x$

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3. Exercises 179

   For the following exercises, find $\frac{dy}{dx}$ for the given functions.

   $y=\frac{\sec x}{x}$

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4. Exercises 181

   For the following exercises, find $\frac{dy}{dx}$ for the given functions.

   $y=(x+\cos x)(1-\sin x)$

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5. Exercises 183

   For the following exercises, find $\frac{dy}{dx}$ for the given functions.

   $y=\frac{1-\cot x}{1+\cot x}$

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6. Exercises 185

   For the following exercises, find an equation of the tangent line to each of the given functions at the indicated values of $x.$ Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.

   [T] $f(x)=\text{-}\sin x,x=0$

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7. Exercises 187

   For the following exercises, find an equation of the tangent line to each of the given functions at the indicated values of $x.$ Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.

   [T] $f(x)=1+\cos x,x=\frac{3\pi }{2}$

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8. Exercises 189

   For the following exercises, find an equation of the tangent line to each of the given functions at the indicated values of $x.$ Then use a calculator to graph both the function and the tangent line to ensure the equation for the tangent line is correct.

   [T] $f(x)=x^2-\tan x,x=0$

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9. Exercises 191

   For the following exercises, find $\frac{d^{2}y}{d{x}^2}$ for the given functions.

   $y=x\sin x-\cos x$

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10. Exercises 193

    For the following exercises, find $\frac{d^{2}y}{d{x}^2}$ for the given functions.

    $y=x-\frac{1}{2}\sin x$

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11. Exercises 195

    For the following exercises, find $\frac{d^{2}y}{d{x}^2}$ for the given functions.

    $y=2\csc x$

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12. Exercises 197

    For the following exercises, find $\frac{d^{2}y}{d{x}^2}$ for the given functions.

    Find all $x$ values on the graph of $f(x)=-3\sin x\cos x$ where the tangent line is horizontal.

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13. Exercises 199

    For the following exercises, find $\frac{d^{2}y}{d{x}^2}$ for the given functions.

    Let $f(x)=\cot x.$ Determine the points on the graph of $f$ for where the tangent line(s) is (are) parallel to the line

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14. Exercises 201

    For the following exercises, find $\frac{d^{2}y}{d{x}^2}$ for the given functions.

    Let the position of a swinging pendulum in simple harmonic motion be given by $s(t)=a\cos t+b\sin t$ where and are constants, measures time in seconds, and measures position in centimeters. If the position is 0 cm and the velocity is 3 cm/s when , find the values of and .

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15. Exercises 203

    For the following exercises, find $\frac{d^{2}y}{d{x}^2}$ for the given functions.

    The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by $y=10+5\sin x$ where is the number of hamburgers sold and represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find and determine the intervals where the number of burgers being sold is increasing.

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16. Exercises 209

    For the following exercises, find the requested higher-order derivative for the given functions.

    $\frac{d^{3}y}{d{x}^3}$ of $y=3\cos x$

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17. Exercises 211

    For the following exercises, find the requested higher-order derivative for the given functions.

    $\frac{d^{4}y}{d{x}^4}$ of $y=5\cos x$

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18. Exercises 213

    For the following exercises, find the requested higher-order derivative for the given functions.

    $\frac{d^{3}y}{d{x}^3}$ of $y=x^{10}-\sec x$

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