Differentiation Quiz

The functions f and g are differentiable for all x and g(x) = f^{-1}(x). If f(x) = x^3 + 2x and f(-1) = -3, what is the value of g^{-3}(-3)?

• -1

• \frac{1}{3}

• 5

• 1

• 5

Tags

CCSS.HSF-BF.B.4A

If f(x) = \sin^{-1}(\ln x), then f'(x) =

• \frac{1}{x(1-\ln x)}

• \frac{1}{x\sqrt{1-\ln^2 x}}

• \frac{1}{x\sqrt{1+\ln^2 x}}

• \frac{1}{x\sqrt{1-\ln^2 x}}

• \frac{-x}{\sqrt{1+\ln^2 x}}

If x - 2xy - y^2 = 2, then when x = 1, \frac{dy}{dx} =

• -\frac{3}{4}

• -\frac{1}{4}

• \frac{1}{4}

• \frac{3}{4}

• nonexistent

For which of the following functions would the quotient rule be considered the best method for finding the derivative?

• \( f(x) = \frac{2x^3 - 4x^2 + 6x}{2x} \)

• \( f(x) = \sin^{-1}(2x) \)

• \( f(x) = \frac{1}{x^2} \)

• \( f(x) = \frac{\sin x}{2x - 1} \)

• \( f(x) = (x^2 - 25)^{-3} \)

The figure above shows the graph of \( f' \), the derivative of \( f \). At which of the following points is \( f''(x) \) the greatest value?

• A

• B

• C

• D

• E

What is the equation of the line tangent to the graph of \( y = \sec(2x) \) at \( x = \frac{\pi}{2} \)?

• \( y = -1 \)

• \( y = x - 1 \)

• \( y = x \)

• \( y = x + 1 \)

• \( y = 1 \)