Question 1

Using the limit definition of the derivative, what is $$f'(x)$$ for $$f(x) = 3x + x^2$?

Accepted Answer

$$f'(x) = 3 + 2x$$

Question 2

Which of the following functions is NOT differentiable at $$x = -3$$?

Accepted Answer

$$f(x) = |x + 3|$$

Question 3

Find the derivative of $$g(x) = e^x \sin(x)$$ using differentiation rules.

Accepted Answer

$$g'(x) = e^x \sin(x) + e^x \cos(x)$$

Question 4

Compute the second derivative $$f''(x)$$ for $$f(x) = \frac{1}{x} + x^4$.

Accepted Answer

$$f''(x) = 2x$$^{-3}$$ + 12x^2$

Question 5

Which of the following best describes the differentiability of $$g(x) = x$$^{1/3}$$ on $$\mathbb{R}$$?

Accepted Answer

$$g(x) is $$differentiable everywhere on $$\mathbb{R}.$$

Question 6

Find the derivative of $$w(x) = x^x$$ for $$x \u003e 0.$$

Accepted Answer

$$w'(x) = x^x (\ln(x) + 1)$$

Question 7

A medication's concentration in the bloodstream is modeled by $$C(t) = e$$^{-t^2}$$, where t$ is time in hours. What is the third derivative C$$'''(t)$$?

Accepted Answer

$$C'''(t) = (12t - 8t^3$) e$$^{-t^2}$$

Question 8

Given $$h(x) = \ln\left(\frac{1-x}{1+x}\right)$$, what is $$h'(x)$$?

Accepted Answer

$$h'(x) = -\frac{2}{1-x^2$}$$

Question 9

Which of the following functions is differentiable everywhere on $$\mathbb{R}$$?

Accepted Answer

$$g(x) = x$$^{1/3}$$

Question 10

Find the derivative of $$v(x) = \frac{\cos(x)}{1 - \sin(x)}.$$

Accepted Answer

$$v'(x) = \frac{-1}{1-\sin(x)}$$

Question 11

A population grows according to $$P(t) = 2t^3$ + e^t - 3\sin(t) + 2\cos(t). $$What is the instantaneous rate of change at $$t = 0$$?

Accepted Answer

$$P'(0) = 1$$

Question 12

Find the derivative of $$u(x) = \frac{1}{\sqrt{1 + x^4$}}.$$

Accepted Answer

$$u'(x) = -\frac{2x^3$}{(1 + x^4$)^{3/2}$$}$

Question 13

Given $$F(x) = \frac{1}{1 + \frac{1}{1 + x}}$$, what is $$F'(x)$$?

Accepted Answer

$$F'(x) = \frac{1}{(1 + x)^2$}$$

Question 14

Find the derivative of $$G(x) = \arcsin(\sqrt{x})$$ for $$x \in [0,1].$$

Accepted Answer

$$G'(x) = \frac{1}{2\sqrt{x}\sqrt{1-x}}$$

Question 15

Suppose $$r(x) = \begin{cases} \sin(x), & x \leq 0 \ xe$$^{-x}$$, & x \u003e 0 \end{cases}. Is$$ $$r(x)$$ differentiable at $$x = 0$$?

Accepted Answer

Yes, because the left and right derivatives at $$x = 0$$ are equal.

Question 16

Find the derivative of $$h(x) = \frac{\ln(x)}{1 + e^x}.$$

Accepted Answer

$$h'(x) = \frac{\frac{1}{x}(1 + e^x) - \ln(x)e^x}{(1 + e^x)^2$}$$

Calculus Practice: Derivatives, Differentiability, and Higher-Order Derivatives

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