Q: Which of the following is the derivative of $$y = \tan(x) \cdot \sec(x)$$?

$$\sec^3$(x) + \tan^2$(x) \sec(x)$$

Question 1

Suppose the position of a car moving along a straight road is given by $$s(t) = 3t^2$ + 2t + 1$$, where $$s is in $$meters and $$t is in $$seconds. What is the instantaneous velocity of the car at $$t = 4$$ seconds?

Accepted Answer

$$26 m/s$$

Question 2

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

Accepted Answer

$$5x^4$$ \cos x - $$x^5$$ \sin x$$

Question 3

A medication's concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$, where t$ is in hours. What is the rate of change of the concentration at t = 2$ hours?

Accepted Answer

$$-5e^{-1}$$

Question 4

Which of the following best describes the chain rule for differentiation?

Accepted Answer

$$[f(g(x))]' = f'(g(x)) \cdot g'(x)$$

Question 5

A ball is thrown upward and its height after $$t$$ seconds is $$h(t) = -16t^2$ + 64t + 80. $$What is the velocity of the ball when it reaches its maximum height?

Accepted Answer

$$0 m/s$$

Question 6

Given $$y = \sin(x^2$)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 7

A company’s profit $$P (in $$thousands of dollars) is given by $$P(x) = 5x^3$ - 2x^2$ + 3x$$, where $$x is $$the number of hundreds of units sold. What is the marginal profit when $$x = 2$$?

Accepted Answer

$$49$$

Question 8

Which of the following is the correct linearization of $$f(x) = \sqrt{x} at$$ $$x = 4$$?

Accepted Answer

$$L(x) = \frac{1}{4}(x - 4) + 2$$

Question 9

If $$f(x) = \frac{\sin x}{x}$$, what is $$f'(x)$$ using the quotient rule?

Accepted Answer

$$\frac{x \cos x - \sin x}{x^2$}$$

Question 10

A chemical reaction’s rate is modeled by $$R(t) = 4t^2$ - 3t + 2. At $$what time $$t$$ does the reaction rate reach a minimum?

Accepted Answer

$$t = \frac{3}{8}$$

Question 11

Which of the following is the derivative of $$y = 	an(x) \cdot \sec(x)$$?

Accepted Answer

$$\sec^3$(x) + 	an^2$(x) \sec(x)$$

Question 12

A projectile is launched and its height is given by $$h(t) = -5t^2$ + 20t + 10. At $$what time does it reach its maximum height?

Accepted Answer

$$t = 2$$

Question 13

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

Accepted Answer

$$4(3x^2$$ + $$2x)^3$$ \cdot (6x + 2)$$

Question 14

Which of the following is the correct application of the product rule to $$y = x^2$ \sin x$$?

Accepted Answer

$$2x \sin x + x^2$ \cos x$$

Question 15

A population grows according to $$P(t) = 1000e$$^{0.03t}$$. What is the instantaneous rate of growth at t = 10$?

Accepted Answer

$$30e^{0.3}$$

Question 16

Which of the following is the correct derivative of $$y = \cot x$$?

Accepted Answer

$$-\csc^2$ x$$

Question 17

If $$f(x) = x^2$ + 2x + 1$$, use the quadratic formula to find the roots.

Accepted Answer

$$x = -1$$

Question 18

Which of the following is the correct derivative of $$y = \sec x$$?

Accepted Answer

$$\sec x 	an x$$

Question 19

A tank is being filled with water such that the volume at time $$t is$$ $$V(t) = t^3$ - 6t^2$ + 9t. At $$what time is the rate of water inflow at a minimum?

Accepted Answer

$$t = 2$$

Question 20

Given $$f(x) = \cos(x^3$)$$, what is $$f'(x)$$?

Accepted Answer

$$-3x^2$$ \$$sin(x^3$$)$$

Essential Derivative Rules and Related Formulas in Calculus I

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