Question 1

Which of the following is the correct derivative of the function $$f(x) = 7$$?

Accepted Answer

$$f'(x) = 0$$

Question 2

Let $$f(x) = x^5$ - 3x^2$ + 2x - 8. $$What is $$f'(x)$$?

Accepted Answer

$$f'(x) = 5x^4$ - 6x + 2$$

Question 3

Suppose $$f(x) = x$$^{-2}$$. What is f$$'(x)$$?

Accepted Answer

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

Question 4

A population of bacteria grows according to $$P(t) = e$$^{2t}$$. What is the rate of growth of the population at time t$?

Accepted Answer

$$P'(t) = 2e$$^{2t}$$

Question 5

Let $$f(x) = 3x^4$ - 2x^2$ + 5x. $$Find the value of $$f''(x).$$

Accepted Answer

$$f''(x) = 12x^2$ - 4$$

Question 6

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

Accepted Answer

$$f'(x) = x^2$ e^x + 2x e^x$$

Question 7

A car's position is given by $$s(t) = 4t^3$ - 3t^2$ + 2t + 1. $$What is the acceleration at $$t = 1$$ second?

Accepted Answer

$$a(1) = 18$$

Question 8

Let $$f(x) = \frac{x^2$ + x - 2}{x^3$ + 6}. $$What is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{(x^3$ + 6)(2x + 1) - (x^2$ + x - 2)(3x^2$)}{(x^3$ + 6)^2$}$$

Question 9

Which of the following statements about the derivative of $$f(x) = e^x is $$true?

Accepted Answer

The derivative is equal to the function itself for all $$x.$$

Question 10

Let $$f(x) = x^4$ e^x. $$What is $$f''(x)$$?

Accepted Answer

$$f''(x) = x^2$(x^2 + 8x + 12)e^x$

Question 11

A medication's concentration in the bloodstream is modeled by $$C(t) = \frac{t}{t^2$ + 1}. $$What is the rate of change of concentration at $$t = 1$$ hour?

Accepted Answer

$$C'(1) = 0$$

Question 12

Let $$f(x) = x^n$$, where $$n is a $$positive integer. Which of the following is the correct formula for $$f'(x)$$?

Accepted Answer

$$f'(x) = n x$$^{n-1}$$

Question 13

Suppose $$f(x) = x^3$ - 3x^2$ + 2. $$What is the third derivative $$f'''(x)$$?

Accepted Answer

$$f'''(x) = 6$$

Question 14

Let $$f(x) = x^2$ - 1$$ and $$g(x) = x^2$ + x + 1. $$What is the slope of the tangent line to $$y = \frac{f(x)}{g(x)} at$$ $$x = 1$$?

Accepted Answer

$$\frac{2}{3}$$

Question 15

If $$f(x) = x e^x$$, what is the $$nth $$derivative $$f^{(n)}(x$$)$$?

Accepted Answer

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

Question 16

Let $$f(x) = \sqrt{x} g(x)$$, where $$g(4) = 2$$ and $$g'(4) = 3. $$What is $$f'(4)$$?

Accepted Answer

$$\frac{13}{2}$$

Question 17

Which of the following best describes the geometric property of the function $$y = e^x at $$the point $$(0, 1)$$?

Accepted Answer

The tangent line at $$(0, 1)$$ has slope $$1.$$

Question 18

Let $$f(x) = x^2$ e^x. $$What is $$f''(x)$$?

Accepted Answer

$$f''(x) = (x^2$ + 4x + 2) e^x$$

Question 19

A function $$y = x^4$ - 6x^2$ + 4$$ has horizontal tangents at which $$x-$$values?

Accepted Answer

$$x = 0,\ x = \sqrt{3},\ x = -\sqrt{3}$$

Differentiation Rules and Applications

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