Question 1

Which of the following is the correct derivative of the function $$f(x) = 7x^5$?

Accepted Answer

$$f'(x) = 35x^4$

Question 2

If $$g(x) = 3x^2$ + 5x + 2$$, what is $$g'(x)$$?

Accepted Answer

$$g'(x) = 6x + 5$$

Question 3

Which rule allows you to find the derivative of $$f(x) = 4x^3$ + 2x by $$differentiating each term separately?

Accepted Answer

The Sum Rule

Question 4

Given $$h(x) = 5e^x$$, what is $$h'(x)$$?

Accepted Answer

$$h'(x) = 5e^x$$

Question 5

A rock is launched vertically upward with a velocity of $$96 ft/s. $$Its position after $$t$$ seconds is $$s(t) = 96t - 16t^2$. What is its velocity after 2$ seconds?

Accepted Answer

$$v(2) = 32 ft/s$$

Question 6

What is the derivative of the constant function $$f(x) = 12$$?

Accepted Answer

$$f'(x) = 0$$

Question 7

If $$f(x) = x^4$ - 2x^2$ + 7$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 4x^3$ - 4x$$

Question 8

Which of the following statements about the derivative of $$e^x is $$true?

Accepted Answer

The derivative of $$e^x is$$ $$e^x$$

Question 9

Find the value of $$\frac{d}{dx}(x^7$).$$

Accepted Answer

$$7x^6$$

Question 10

If $$f(x) = 2x^3$ + 4x^2$, at which x$ does f(x$$)$$ have a horizontal tangent?

Accepted Answer

$$x = 0$$

Question 11

Given $$f(x) = x^2$ and g(x) = 3x$, what is $$\frac{d}{dx}(f(x) + g(x))$$?

Accepted Answer

$$2x + 3$$

Question 12

Which of the following is the correct application of the constant multiple rule for derivatives?

Accepted Answer

$$\frac{d}{dx}(7x^2$) = 7 \cdot 2x = 14x$$

Question 13

A function $$f(x)$$ has a tangent line at $$x = 1$$ with equation $$y = 9x - 6. If$$ $$f(1) = 3$$, what is $$f'(1)$$?

Accepted Answer

$$f'(1) = 9$$

Question 14

If $$f(x) = x^3$ + 2x^2$ + x$$, what is the second derivative $$f''(x)$$?

Accepted Answer

$$f''(x) = 6x + 4$$

Question 15

Which of the following is the correct derivative of $$f(x) = \sqrt{x}$$?

Accepted Answer

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

Question 16

If $$f(x) = x^2$ and g(x) = e^x$, what is $$\frac{d}{dx}(f(x) + g(x))$$?

Accepted Answer

$$2x + e^x$$

Question 17

A rock is launched upward with position $$s(t) = 96t - 16t^2$. What is its acceleration at any time t$?

Accepted Answer

$$a(t) = -32$$

Question 18

Which of the following is the correct derivative of $$f(x) = x^n$$ for any real number $$n$$?

Accepted Answer

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

Question 19

If $$f(x) = 3x^2$ + 4e^x$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 6x + 4e^x$$

Derivatives of Polynomials & Exponentials: Rules, Examples, and Applications

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