Question 1

Which of the following correctly states the derivative of the function $$f(x) = e^x$$?

Accepted Answer

$$\frac{d}{dx}(e^x) = e^x$$

Question 2

Suppose the population of a bacteria culture at time $$t (in $$hours) is modeled by $$P(t) = e^t. $$What is the instantaneous rate of change of the population at $$t = 2$$ hours?

Accepted Answer

$$e^2$$

Question 3

If $$f(x) = 5e^x$$, what is $$f'(x)$$?

Accepted Answer

$$5e^x$$

Question 4

Which rule allows you to differentiate $$f(x) = 7e^x + 3 as$$ $$f'(x) = 7e^x$$?

Accepted Answer

The Sum Rule and the Constant Multiple Rule

Question 5

Given $$f(x) = e^x + x^2$, what is f$$''(x)$$?

Accepted Answer

$$e^x + 2x$$

Question 6

If $$y = e^x$$, what is the third derivative $$y'''$$?

Accepted Answer

$$e^x$$

Question 7

A medication's concentration in the bloodstream is modeled by $$C(t) = 2e^t. $$What is the rate of change of concentration at any time $$t$$?

Accepted Answer

$$2e^t$$

Question 8

Which of the following is NOT a correct application of the derivative rule for $$e^x$$?

Accepted Answer

$$\frac{d}{dx}(e^x) = x e$$^{x-1}$$

Question 9

If $$f(x) = e^x + 4x$$, what is $$f'(x)$$?

Accepted Answer

$$e^x + 4$$

Question 10

Which of the following best describes why $$e^x is $$unique among exponential functions in calculus?

Accepted Answer

Its derivative is itself: $$\frac{d}{dx}(e^x) = e^x$$

Question 11

Find the second derivative of $$f(x) = 3e^x - x.$$

Accepted Answer

$$3e^x$$

Question 12

If $$f(x) = e^x$$, which of the following statements is true for all $$n \geq 1$$?

Accepted Answer

$$f^{(n)}(x) = e^x$

Question 13

A radioactive substance decays according to $$N(t) = e$$^{-t}$$. What is N$$'(t)$$?

Accepted Answer

$$-e^{-t}$$

Question 14

Which of the following is the correct application of the sum rule for differentiation?

Accepted Answer

$$\frac{d}{dx}(e^x + x^2$) = e^x + 2x$$

Question 15

If $$f(x) = e^x$$, what is the value of $$f'(0)$$?

Accepted Answer

$$1$$

Rules of Differentiation and Higher Order Derivatives

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