Q: Given $$f(x) = \tan(x) \cdot x^2$, find f$$'(x)$$.

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

Question 1

Let $$f(x) = x^3$ e^x. $$What is $$f'(x)$$?

Accepted Answer

$$x^3 e^x$$ + $$3x^2 e^x$

Question 2

Suppose $$y is $$defined implicitly by $$x^2$$ + $$y^2 = 16$. What is $$\frac{dy}{dx}$$ at the point (3$$,1)$$?

Accepted Answer

$$-\frac{3}{1}$$

Question 3

A stone is thrown off a cliff and its height above the ground at time $$t$$ seconds is given by $$s(t) = -16t^2$ + 96. $$What is the velocity when it hits the ground?

Accepted Answer

$$-80 m/s$$

Question 4

Given $$f(x) = 	an(x) \cdot x^2$, find f$$'(x)$$.

Accepted Answer

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

Question 5

Suppose oil is spilled from a tanker and spreads in a circle whose area increases at a constant rate of $$6 mi$^2/hr. $$How fast is the radius increasing when the radius is $$2 mi$$?

Accepted Answer

$$\frac{6}{4\pi} mi/hr$$

Question 6

Let $$y = \sin(x) e$$^{x^2}$$. Find $$\frac{dy}{dx}$$.

Accepted Answer

$$\cos(x) e$$^{x^2}$$ + 2x \sin(x) e$$^{x^2}$$

Question 7

A balloon in the shape of a sphere is being inflated so that its volume increases at $$300 mm$^3/sec. $$How fast is the diameter increasing when the radius is $$5 mm$$?

Accepted Answer

$$\frac{3}{25\pi} mm/sec$$

Question 8

Find the equation of the tangent line to $$y = \sin(x) at$$ $$x = \frac{\pi}{4}.$$

Accepted Answer

$$y = \frac{\sqrt{2}}{2} (x - \frac{\pi}{4}) + \frac{\sqrt{2}}{2}$$

Question 9

Use logarithmic differentiation to find $$\frac{dy}{dx}$$ for $$y = x^x.$$

Accepted Answer

$$x^x (1 + \ln(x))$$

Question 10

Given $$f(x) = \frac{\sin(x)}{x}$$, what is $$f'(x)$$?

Accepted Answer

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

Question 11

Suppose a person is standing $$40 m $$east of an intersection while a car drives north through the intersection at $$30 m/s. At $$the moment the car is $$30 m $$north of the intersection, how fast is the distance between the person and the car changing?

Accepted Answer

$$24 m/s$$

Question 12

Find the third derivative of $$f(x) = 4e$$^{5x}$$.

Accepted Answer

$$500e^{5x}$$

Question 13

If $$f(x) = \ln(\sin(x))$$, what is $$f'(x)$$?

Accepted Answer

$$\cot(x)$$

Question 14

Given $$f(x) = \sec(x) 	an(x)$$, what is $$f'(x)$$?

Accepted Answer

$$\sec^3$(x)$$

Question 15

Let $$f(x) = \ln(x^2$ + 1). $$What is $$f'(x)$$?

Accepted Answer

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

Question 16

If $$y is $$defined implicitly by $$x^2 y$$ + $$y^2 = 7$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 17

Find the derivative of $$f(x) = \ln(3x^2$).$$

Accepted Answer

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

Question 18

If $$f(x) = \frac{1}{x^2$}$$, what is $$f'(x)$$?

Accepted Answer

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

Question 19

Let $$f(x) = e$$^{x}$$ \cos(x). $$What is $$f'(x)$$?

Accepted Answer

$$e^{x}$$ \cos(x) - $$e^{x}$$ \sin(x)$$

Question 20

Suppose $$f(x) = \ln(x) \cdot x^2$. What is f$$'(x)$$?

Accepted Answer

$$2x \ln(x) + x$$

Exam 2 Calculus Review – Step-by-Step Derivative and Related Rates Guidance

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