Question 1

Which of the following is the correct derivative of $$y = \log_b g(x)$$ with respect to $$x$$?

Accepted Answer

$$y' = \dfrac{g'(x)}{g(x) \ln b}$$

Question 2

Given $$y = \ln(2x^4$ + x)$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{8x^3$ + 1}{2x^4$ + x}$$

Question 3

If $$f(x) = \log_5$(4 + \cos x^3$)$$, what is $$f'(x)$$?

Accepted Answer

$$\dfrac{-2x \sin x^3$}{(4 + \cos x^3$) \ln 5}$$

Question 4

Find $$\dfrac{d}{dx} \left( \ln \dfrac{x+1}{\sqrt{x-2}} \right).$$

Accepted Answer

$$\dfrac{x+1}{x-2} - \dfrac{1}{2} \cdot \dfrac{x+1}{(x-2$$)^{3/2}$$}$$

Question 5

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

Accepted Answer

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

Question 6

Given $$y = 2(\sin$$^{-1}$$ x)^2$, find $$\dfrac{dy}{dx}$$ using the chain rule.

Accepted Answer

$$\dfrac{4 \sin$$^{-1}$$ x}{\sqrt{1-x^2$}}$$

Question 7

Find $$\dfrac{d}{dx} \left( \cos$$^{-1}$$(x^3 + 2$$) \right)$.

Accepted Answer

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

Question 8

Which formula gives the derivative of $$y = 	an$$^{-1}$$(g(x))$$?

Accepted Answer

$$\dfrac{g'(x)}{1 + (g(x))^2$}$$

Question 9

Suppose $$y = \sec$$^{-1}$$(f(x)). $$What is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{f'(x)}{f(x) \sqrt{f(x)^2$ - 1}}$$

Question 10

For the equation $$x^2$$ + $$y^2 = 4$, which method should be used to find $$\dfrac{dy}{dx}$$?

Accepted Answer

Implicit differentiation

Question 11

Using implicit differentiation, find $$\dfrac{dy}{dx}$$ for $$x^2$$ + $$y^2 = 4$.

Accepted Answer

$$\dfrac{-x}{y}$$

Question 12

Given $$x^2$$ + $$4xy^2$$ + $$y^2 = 4$, use implicit differentiation to find $$\dfrac{dy}{dx}$$.

Accepted Answer

$$\dfrac{-2x - 4y}{4x y + 2y}$$

Question 13

For the folium of Descartes $$x^3$$ + $$y^3 = 6xy$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$\dfrac{3x^2$ - 6y}{6x - 3y^2$}$$

Question 14

At the point $$(3,3) on $$the folium $$x^3$$ + $$y^3 = 6xy$, what is the slope of the tangent line?

Accepted Answer

$$0$$

Question 15

For which value of $$y is $$the tangent to the folium $$x^3$$ + $$y^3 = 6xy$ horizontal in the first quadrant?

Accepted Answer

$$y = \dfrac{1}{2}x^2$

Question 16

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

Accepted Answer

$$\dfrac{-g'(x)}{1 + (g(x))^2$}$$

Question 17

If $$y = \csc(x^2$)$$, what is $$\dfrac{dy}{dx}$$?

Accepted Answer

$$-\csc(x^2$) \cot(x^2$) \cdot 2x$$

Question 18

Which rule is most appropriate for differentiating $$y = \ln(g(x))$$?

Accepted Answer

Chain rule

Derivatives of Logarithmic, Inverse Trigonometric Functions, and Implicit Differentiation

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