$$P'(10) = 100 \times 0.03 \times e$$^{0.3}$$

Question 1

Which of the following is the correct derivative of the function $$f(x) = 7x^5$ - 3x^2$ + 4$$?

Accepted Answer

$$f'(x) = 35x^4$ - 6x$$

Question 2

A medication's concentration in the bloodstream is modeled by $$C(t) = 5e^t - 2t^2$. What is the rate of change of concentration at t = 0$?

Accepted Answer

$$C'(0) = 5e^0$ - 4(0) = 5$$

Question 3

Which rule must be used to differentiate $$y = (3x^2$ + 2x)(x^3$ - 1)$$?

Accepted Answer

Product Rule

Question 4

Find the derivative of $$f(x) = \frac{2x^3$ + 1}{x^2$}.$$

Accepted Answer

$$f'(x) = \frac{(x^2$)(6x^2$$) - $$(2x^3 + 1)(2x$$)}{$$x^4$$}$

Question 5

A population grows according to $$P(t) = 100e$$^{0.03t}$$. What is the instantaneous rate of growth at t = 10$?

Accepted Answer

$$P'(10) = 100 	imes 0.03 	imes e$$^{0.3}$$

Question 6

Which of the following statements about the derivative of a constant function is correct?

Accepted Answer

The derivative is always zero.

Question 7

Given $$y = \frac{6e^x}{1 + e^x}$$, what is $$y'$$?

Accepted Answer

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

Question 8

Find the equation of the tangent line to $$y = x^3$ - 3x - 2 at $$the point where the tangent is horizontal.

Accepted Answer

$$y = -4$$

Question 9

Which of the following is a common misconception about the product rule?

Accepted Answer

That $$\frac{d}{dx}[f(x)g(x)] = f'(x)g'(x)$$

Question 10

If $$f(x) = 6e^x - 2x^2$ + 3$$, what is $$f''(x)$$?

Accepted Answer

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

Question 11

A car's position is given by $$s(t) = t^3$ - 6t^2$ + 9t + 2. $$What is the car's acceleration at $$t = 2$$?

Accepted Answer

$$a(2) = 6t - 12$$, so $$a(2) = 0$$

Question 12

Which rule is used to differentiate $$y = \frac{x^2$ e^x}{x^3$ + 5e^x}$$?

Accepted Answer

Quotient Rule (with Product Rule in numerator)

Question 13

Find the derivative of $$h(x) = 9 + 8x - 6x^2$.

Accepted Answer

$$h'(x) = 8 - 12x$$

Question 14

A chemical reaction's rate is modeled by $$R(t) = 4e^t t^3$. What is R$$'(t)$$?

Accepted Answer

$$R'(t) = 4e^t t^3$ + 12e^t t^2$

Question 15

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

Accepted Answer

$$\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$$

Question 16

Find the equation of the tangent line to $$y = (5x - 1)(2x^2$ + x) at$$ $$x = 1.$$

Accepted Answer

$$y = 35x - 23$$

Question 17

Which of the following is the correct second derivative notation?

Accepted Answer

$$\frac{d^2$y}{dx^2$}$$

Question 18

If $$f(x) = x$$^{3/2}$$ + x$$^{-3}$$, what is f$$'(x)$$?

Accepted Answer

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

Differentiation Rules and Applications in Calculus

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