Q: Which of the following is the correct derivative of $$f(x) = \sec(5x + \pi/2)$$?

$$f'(x) = 5\sec(5x + \pi/2)\tan(5x + \pi/2)$$

Q: Evaluate $$\int \left[\cos(\pi x) + \sec^2$\left(\dfrac{x}{3} + 1\right) + e$$^{5x-2}$$ + (4x + 5)^5$\right] dx.$$

$$\dfrac{1}{\pi}\sin(\pi x) + 3\tan\left(\dfrac{x}{3} + 1\right) + \dfrac{1}{5}e$$^{5x-2}$$ + \dfrac{1}{24}(4x + 5)^6$ + C$$

Question 1

Which of the following correctly applies the chain rule to find the derivative of $$f(x) = (2x + 3)^6$?

Accepted Answer

$$f'(x) = 12(2x + 3)^5$

Question 2

A pharmaceutical company models the concentration of a drug in the bloodstream as $$C(t) = e$$^{0.2t}$$, where t$ is in hours. What is the rate of change of the drug concentration at any time t$?

Accepted Answer

$$C'(t) = 0.2e$$^{0.2t}$$

Question 3

Let $$f(x) = \arctan(x). $$What is the second derivative $$f''(x)$$?

Accepted Answer

$$f''(x) = -\dfrac{2x}{(1 + x^2$)^2$$}$

Question 4

Suppose $$f(x) = \sqrt{x}$$ models the radius (in cm) of a tumor as a function of time $$x (in $$weeks). What is the equation of the tangent line to $$f(x) at$$ $$x_0 = 9$?

Accepted Answer

$$L(x) = 3 + \dfrac{1}{6}(x - 9)$$

Question 5

Evaluate the following indefinite integral: $$\int (x^2$ + x$$^{-5}$$ + \sqrt{x} + x$$^{2/3}$$)\,dx.$$

Accepted Answer

$$\dfrac{1}{3}x^3$ - \dfrac{1}{4}x$$^{-4}$$ + \dfrac{2}{3}x$$^{3/2}$$ + \dfrac{3}{5}x$$^{5/3}$$ + C$$

Question 6

A population grows according to $$P(t) = 5t^2$ - 4t + 5t$$^{-4}$$ for t$$ \geq 0$$. What is P$$'(t)$$?

Accepted Answer

$$P'(t) = 10t - 4 - 20t$$^{-5}$$

Question 7

Which of the following is the correct derivative of $$f(x) = \sec(5x + \pi/2)$$?

Accepted Answer

$$f'(x) = 5\sec(5x + \pi/2)	an(5x + \pi/2)$$

Question 8

Find the derivative of $$f(x) = (3x + 5)^5$ - (x^2$ + 4$$)^{-3}$$ raised to the 9th power, i.e., f(x$$) = \left[(3x + $$5)^5$$ - $$(x^2$$ + $$4)^{-3}$$\right]^9$$.

Accepted Answer

$$f'(x) = 9\left[(3x + 5)^5$ - (x^2$ + 4$$)^{-3}$$\right]^8 \left[15(3x + 5)^4$ + 3(2x)(x^2$ + 4$$)^{-4}$$\right]$$

Question 9

A car's velocity is given by $$v(t) = \sin(t)$$ meters per second. What is the acceleration at time $$t$$?

Accepted Answer

$$a(t) = \cos(t)$$

Question 10

Evaluate $$\int \left[\cos(\pi x) + \sec^2$\left(\dfrac{x}{3} + 1\right) + e$$^{5x-2}$$ + (4x + 5)^5$\right] dx.$$

Accepted Answer

$$\dfrac{1}{\pi}\sin(\pi x) + 3	an\left(\dfrac{x}{3} + 1ight) + \dfrac{1}{5}e$$^{5x-2}$$ + \dfrac{1}{24}(4x + 5)^6$ + C$$

Question 11

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

Accepted Answer

$$f'(x) = \dfrac{1}{x}$$

Question 12

Let $$f(x) = x^4$ + 5x^3$ - 4x^2$ + 6x - 1. $$What is $$f^{(3)}(x$$)$$, the third derivative?

Accepted Answer

$$f^{(3)}(x) = 24x + 30$

Question 13

A chemical reaction's rate is modeled by $$r(x) = 	an(x). $$What is the instantaneous rate of change of the rate at $$x$$?

Accepted Answer

$$r'(x) = \sec^2$(x)$$

Question 14

Evaluate $$\int \dfrac{x + 5}{x - 2} dx.$$

Accepted Answer

$$x + 7\ln|x - 2| + C$$

Question 15

Which of the following is the correct derivative of $$f(x) = \arcsin(x)$$?

Accepted Answer

$$f'(x) = \dfrac{1}{\sqrt{1 - x^2$}}$$

Question 16

A function $$f is $$the derivative of $$g if $$and only if:

Accepted Answer

$$\dfrac{d}{dx}g(x) = f(x)$$

Question 17

Evaluate $$\int \dfrac{x^2$ + 11x + 6}{x^2$} dx.$$

Accepted Answer

$$x + 11\ln|x| - 6x$$^{-1}$$ + C$$

Question 18

Let $$f(x) = \sin(x) + \cos(x). $$What is $$f'(x)$$?

Accepted Answer

$$f'(x) = \cos(x) - \sin(x)$$

Essential Derivative and Integration Rules with Applications

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