Q: If $f(x) = x \tan^{-1}\left(\dfrac{x}{2}\right)$, what is $f'(x)$?

$\tan^{-1}\left(\dfrac{x}{2}\right) + \dfrac{x}{1 + (x/2)^2} \cdot \dfrac{1}{2}$

Question 1

Which of the following is the correct derivative of $y = \sin^{-1}(x)$ with respect to $x$?

Accepted Answer

$\dfrac{1}{\sqrt{1-x^2}}$

Question 2

Find the derivative of $y = 	an^{-1}(e^{3x})$ with respect to $x$.

Accepted Answer

$\dfrac{3e^{3x}}{1+e^{6x}}$

Question 3

A medical imaging device uses the function $y = \cos^{-1}(t^2)$ to model the angle of a rotating sensor, where $t$ is time in seconds. What is $\dfrac{dy}{dt}$?

Accepted Answer

$-\dfrac{2t}{\sqrt{1-t^4}}$

Question 4

Which of the following is the correct derivative of $y = \sec^{-1}(x)$ with respect to $x$?

Accepted Answer

$\dfrac{1}{|x|\sqrt{x^2-1}}$

Question 5

If $y = 3x \sin^{-1}x$, what is $\dfrac{dy}{dx}$?

Accepted Answer

$3\sin^{-1}x + \dfrac{3x}{\sqrt{1-x^2}}$

Question 6

A chemist models the concentration of a substance as $y = \sin^{-1}(8x^4)$. What is $\dfrac{dy}{dx}$?

Accepted Answer

$\dfrac{32x^3}{\sqrt{1-64x^8}}$

Question 7

Find the derivative of $y = 2\cos^{-1}t \cdot \cos^{-1}t$ with respect to $t$.

Accepted Answer

$\dfrac{2\cos^{-1}t}{\sqrt{1-t^2}}$

Question 8

If $f(x) = x 	an^{-1}\left(\dfrac{x}{2}ight)$, what is $f'(x)$?

Accepted Answer

$	an^{-1}\left(\dfrac{x}{2}ight) + \dfrac{x}{1 + (x/2)^2} \cdot \dfrac{1}{2}$

Question 9

Evaluate $f'(2)$ for $f(x) = x 	an^{-1}\left(\dfrac{x}{2}ight)$.

Accepted Answer

$\dfrac{\pi}{4} + \dfrac{1}{2}$

Question 10

A satellite dish is oriented according to $g(x) = \sec^{-1}(2x)$. Find the slope of the tangent line to $g(x)$ at $x=1$.

Accepted Answer

$\dfrac{\sqrt{3}}{3}$

Question 11

Find the equation of the tangent line to $g(x) = \sec^{-1}(2x)$ at the point $(1, \dfrac{\pi}{3})$.

Question 12

Which of the following is the correct derivative of $y = \cot^{-1}(x)$ with respect to $x$?

Accepted Answer

$-\dfrac{1}{1+x^2}$

Question 13

A physicist uses $y = \csc^{-1}(x)$ to model the angle of a pendulum. What is $\dfrac{dy}{dx}$?

Accepted Answer

$-\dfrac{1}{|x|\sqrt{x^2-1}}$

Question 14

Which of the following is the correct derivative of $y = 	an^{-1}(x)$ with respect to $x$?

Accepted Answer

$\dfrac{1}{1+x^2}$

Question 15

If $y = \sin^{-1}(u)$, where $u = 8x^4$, what is $\dfrac{dy}{dx}$ using the chain rule?

Accepted Answer

$\dfrac{1}{\sqrt{1-u^2}} \cdot \dfrac{du}{dx}$

Question 16

A biologist models the growth of a population with $y = 	an^{-1}(e^{x})$. What is $\dfrac{dy}{dx}$?

Accepted Answer

$\dfrac{e^{x}}{1 + e^{2x}}$

Question 17

Which of the following is the correct derivative of $y = \cos^{-1}(x)$ with respect to $x$?

Accepted Answer

$-\dfrac{1}{\sqrt{1-x^2}}$

Derivatives of Inverse Trigonometric Functions 3.10

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