Question 1

A biologist models the population of bacteria in a petri dish as $$P(t) = e$$^{2t^2 + 3t}$$, where t$ is time in hours. What is the rate of change of the population at time t$?

Accepted Answer

$$e^{2t^2 + 3t}(4t + 3$$)$$

Question 2

Find the derivative of $$y = \ln(4 + \cos x)$$ with respect to $$x.$$

Accepted Answer

$$\frac{-\sin x}{4 + \cos x}$$

Question 3

A medication's concentration in the bloodstream is modeled by $$C(x) = \sqrt{x^2$ + 2x + 3}. $$What is $$\frac{dC}{dx}$$?

Accepted Answer

$$\frac{2x + 2}{2\sqrt{x^2$ + 2x + 3}}$$

Question 4

Which of the following is the correct derivative of $$y = \sin(\ln x)$$?

Accepted Answer

$$\frac{\cos(\ln x)}{x}$$

Question 5

A physicist models the displacement of a spring as $$s(t) = \cos(3t). $$What is the acceleration $$a(t)$$, the second derivative of $$s(t)$$?

Accepted Answer

$$-9\cos(3t)$$

Question 6

Find the derivative of $$y = (x^2$ + 3$$)^{2/3}$$.

Accepted Answer

$$\frac{4x}{3}(x^2$ + 3$$)^{-1/3}$$

Question 7

A chemical reaction rate is given by $$R(x) = \ln(x + 2)^2$. What is R$$'(x)$$?

Accepted Answer

$$\frac{2}{x + 2}$$

Question 8

Find the derivative of $$y = \cot$$^{-1}$$(x)$$ with respect to $$x.$$

Accepted Answer

$$\frac{-1}{1 + x^2$}$$

Question 9

A population model is $$P(x) = \ln(10x + 5). $$What is $$P'(x)$$?

Accepted Answer

$$\frac{10}{10x + 5}$$

Question 10

Find the derivative of $$y = \sec(2x)$$ with respect to $$x.$$

Accepted Answer

$$2\sec(2x)	an(2x)$$

Question 11

A function is given by $$f(x) = (2x^2$ + x)^3$. What is f$$'(x)$$?

Accepted Answer

$$3(2x^2$$ + $$x)^2(4x + 1$$)$$

Question 12

Find the derivative of $$y = \sin$$^{-1}$$(x)$$ with respect to $$x.$$

Accepted Answer

$$\frac{1}{\sqrt{1 - x^2$}}$$

Question 13

A sound wave is modeled by $$y = \sin(9x^2$). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

$$18x \cos(9x^2$)$$

Question 14

Find the derivative of $$y = \ln(x^2$ + 1)$$ with respect to $$x.$$

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 15

A function is given by $$f(x) = 	an$$^{-1}$$(x). $$What is $$f'(x)$$?

Accepted Answer

$$\frac{1}{1 + x^2$}$$

Question 16

Find the derivative of $$y = \cos(7x)$$ with respect to $$x.$$

Accepted Answer

$$-7\sin(7x)$$

Question 17

A function is given by $$f(x) = \ln(x + 2)^2$. What is f$$'(x)$$?

Accepted Answer

$$\frac{2}{x + 2}$$

Question 18

Find the derivative of $$y = (x + 1$$)^{-2}$$ with respect to x$.

Accepted Answer

$$-2(x + 1$$)^{-3}$$

Question 19

A function is given by $$f(x) = \sec$$^{-1}$$(x). $$What is $$f'(x)$$?

Accepted Answer

$$\frac{1}{|x|\sqrt{x^2$ - 1}}$$

Differentiation Rules and Applications in Calculus

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