Question 1

Which of the following is the correct derivative of the function $$f(x) = 7x^5$ - 3x + 2$$?

Accepted Answer

$$35x^4 - 3$

Question 2

A pharmaceutical company models the concentration of a drug in the bloodstream as $$C(t) = 12e^t - 5. $$What is the rate of change of the drug concentration with respect to time?

Accepted Answer

$$12e^t$$

Question 3

Find the derivative of $$f(x) = 6\sqrt{x}.$$

Accepted Answer

$$3x^{-1/2}$$

Question 4

Which rule allows you to differentiate $$f(x) = 4x^3$ + 7x as$$ $$f'(x) = 12x^2$ + 7$$?

Accepted Answer

Sum Rule

Question 5

If $$f(x) = 5x$$^{-3}$$ + 4x$$^{-1}$$, what is f$$'(x)$$?

Accepted Answer

$$-15x^{-4}$$ - $$4x^{-2}$$

Question 6

A company’s profit is modeled by $$P(x) = 12x^5$ - 40x + e^x - e^4$. What is the marginal profit function?

Accepted Answer

$$60x^4 - 40 + e^x$

Question 7

For what value of $$x is $$the tangent to $$f(x) = 6x^2$ - 8x$$ equal to $$16$$?

Accepted Answer

$$x = 2$$

Question 8

Find all $$x$$ where the tangent line to $$f(x) = x^3$ + 4x^2$ - 3x is $$horizontal.

Accepted Answer

$$x = -3, \ \frac{1}{3}$$

Question 9

What is the equation of the tangent line to $$f(x) = x^3$ + 4x^2$ - 3x at $$the point $$(1, 2)$$?

Accepted Answer

$$y = 8x - 6$$

Question 10

Which of the following best describes the second derivative $$f''(x)$$?

Accepted Answer

The derivative of the first derivative $$f'(x)$$

Question 11

Find the first and second derivatives of $$f(x) = 2x^3$ + 9x^2$ + 6x.$$

Accepted Answer

$$f'(x) = 6x^2$ + 18x + 6, \ f''(x) = 12x + 18$$

Question 12

Which rule is used to differentiate $$f(x) = c \cdot x^n$$?

Accepted Answer

Constant Multiple Rule

Question 13

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

Accepted Answer

$$e^x$$

Question 14

A population grows according to $$P(t) = 100e$$^{0.03t}$$. What is the instantaneous rate of growth at time t$?

Accepted Answer

Question 15

Which of the following is the correct notation for the $$nth $$derivative of $$f(x)$$?

Accepted Answer

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

Rules of Differentiation: Calculus Study Notes

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