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 $$(1, \dfrac{\pi}{3}).$$

Accepted Answer

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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