Calculus Volume 1 · 3.7 Derivatives of Inverse Functions

Derivatives of Inverse Functions: practice problems

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

1. Exercises 261

   For the following exercises, use the graph of $y=f(x)$ to

   1. sketch the graph of $y=f^{-1}(x),$ and
   2. use part a. to estimate ${(f^{-1})}^{\prime }(1).$

   

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2. Exercises 263

   For the following exercises, use the graph of $y=f(x)$ to

   1. sketch the graph of $y=f^{-1}(x),$ and
   2. use part a. to estimate ${(f^{-1})}^{\prime }(1).$

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3. Exercises 265

   For the following exercises, use the functions $y=f(x)$ to find

   1. $\frac{df}{dx}$ at $x=a$ and
   2. $x=f^{-1}(y).$

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4. Exercises 267

   For the following exercises, use the functions $y=f(x)$ to find

   1. $\frac{df}{dx}$ at $x=a$ and
   2. $x=f^{-1}(y).$

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5. Exercises 269

   For each of the following functions, find ${(f^{-1})}^{\prime }(a).$

   $f(x)=x^3+2x+3,a=0$

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6. Exercises 271

   For each of the following functions, find ${(f^{-1})}^{\prime }(a).$

   $f(x)=x-\frac{2}{x},x<0,a=1$

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7. Exercises 273

   For each of the following functions, find ${(f^{-1})}^{\prime }(a).$

   $f(x)=\tan x+3x^2,a=0;0<x<\pi /2$

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8. Exercises 275

   For each of the given functions $y=f(x),$

   1. find the slope of the tangent line to its inverse function $f^{-1}$ at the indicated point $P,$ and
   2. find an equation of the tangent line to the graph of $f^{-1}$ at the indicated point.

   $f(x)=\sqrt{x-4},P(2,8)$

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9. Exercises 277

   For each of the given functions $y=f(x),$

   1. find the slope of the tangent line to its inverse function $f^{-1}$ at the indicated point $P,$ and
   2. find an equation of the tangent line to the graph of $f^{-1}$ at the indicated point.

   $f(x)=\text{-}{x}^3-x+2,P(-8,2)$

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10. Exercises 279

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

    $y={\sin }^{-1}(x^2)$

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11. Exercises 281

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

    $y={\sec }^{-1}(\frac{1}{x})$

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12. Exercises 282

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

    $y=\sqrt{{\csc }^{-1}x}$

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13. Exercises 283

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

    $y={(1+{\tan }^{-1}x)}^3$

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14. Exercises 285

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

    $y=\frac{1}{{\tan }^{-1}(x)}$

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15. Exercises 287

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

    $y={\cot }^{-1}\sqrt{4-x^2}$

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16. Exercises 289

    For the following exercises, use the given values to find ${(f^{-1})}^{\prime }(a).$

    $f(\pi )=0,f^{\prime }(\pi )=-1,a=0$

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17. Exercises 291

    For the following exercises, use the given values to find ${(f^{-1})}^{\prime }(a).$

    $f(\frac{1}{3})=-8,f^{\prime }(\frac{1}{3})=2,a=-8$

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18. Exercises 293

    For the following exercises, use the given values to find ${(f^{-1})}^{\prime }(a).$

    $f(1)=-3,f^{\prime }(1)=10,a=-3$

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19. Exercises 295

    For the following exercises, use the given values to find ${(f^{-1})}^{\prime }(a).$

    [T] The position of a moving hockey puck after $t$ seconds is $s(t)={\tan }^{-1}t$ where $s$ is in meters.

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20. Exercises 297

    For the following exercises, use the given values to find ${(f^{-1})}^{\prime }(a).$

    [T] A pole stands 75 feet tall. An angle $\theta$ is formed when wires of various lengths of $x$ feet are attached from the ground to the top of the pole, as shown in the following figure. Find the rate of change of the angle $\frac{d\theta }{dx}$ when a wire of length 90 feet is attached.

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21. Exercises 298

    For the following exercises, use the given values to find ${(f^{-1})}^{\prime }(a).$

    [T] A television camera at ground level is 2000 feet away from the launching pad of a space rocket that is set to take off vertically, as seen in the following figure. The angle of elevation of the camera can be found by $\theta ={\tan }^{-1}(\frac{x}{2000}),$ where is the height of the rocket. Find the rate of change of the angle of elevation after launch when the camera and the rocket are 5000 feet apart.

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22. Exercises 299

    For the following exercises, use the given values to find ${(f^{-1})}^{\prime }(a).$

    [T] A local movie theater with a 30-foot-high screen that is 10 feet above a person’s eye level when seated has a viewing angle $\theta$ (in radians) given by $\theta ={\cot }^{-1}\frac{x}{40}-{\cot }^{-1}\frac{x}{10},$

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23. Review Exercises 373

    Find the derivatives of the following functions.

    $f(x)=3x^3-\frac{4}{x^2}$

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24. Review Exercises 375

    Find the derivatives of the following functions.

    $f(x)=e^{\sin x}$

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25. Review Exercises 377

    Find the derivatives of the following functions.

    $f(x)=x^2\cos x+x\tan (x)$

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26. Review Exercises 379

    Find the derivatives of the following functions.

    $f(x)=\frac{x}{4}{\sin }^{-1}(x)$

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