Q: Which of the following is the correct limit definition of the derivative $$f'(a)$$?

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Question 1

Which of the following best describes the derivative of a function $$f at a $$point $$x = a$$?

Accepted Answer

The slope of the tangent line to the graph of $$f at$$ $$x = a.$$

Question 2

Given $$f(x) = x^2$ + 3x$$, use the definition of the derivative to find $$f'(2). $$Which of the following is correct?

Accepted Answer

$$f'(2) = 7$$

Question 3

Which of the following is the correct limit definition of the derivative $$f'(a)$$?

Accepted Answer

$$f'(a) = \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 4

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

Accepted Answer

$$\cos(x)$$

Question 5

Which rule is used to differentiate $$f(x) = x^2$ e^x$$?

Accepted Answer

Product Rule

Question 6

Find the derivative of $$f(x) = \ln(x^2$ + 1).$$

Accepted Answer

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

Question 7

A balloon is rising vertically at a rate of $$4 m/s. At $$what rate is the distance from a point $$60 m $$from the ground station increasing when the balloon is $$80 m $$high?

Accepted Answer

$$\frac{16}{5} m/s$$

Question 8

If $$y = e$$^{2x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2e^{2x}$$

Question 9

Which of the following statements is true about differentiability?

Accepted Answer

If a function is differentiable at $$x = a$$, then it is continuous at $$x = a.$$

Question 10

Find the equation of the tangent line to $$f(x) = x^3$ - x at$$ $$x = 2.$$

Accepted Answer

$$y = 11x - 20$$

Question 11

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

Accepted Answer

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

Question 12

Which of the following is the derivative of $$f(x) = 	an(x)$$?

Accepted Answer

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

Question 13

If $$f(x) = x^4$ - 2x^2$ + 1$$, at which $$x$$ does the graph of $$f$$ have a horizontal tangent line?

Accepted Answer

$$x = 0, x = 1, x = -1$$

Question 14

Find $$\frac{d}{dx}(\sin(x) \cos(x)).$$

Accepted Answer

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

Question 15

Which of the following is the correct application of the Chain Rule?

Accepted Answer

To differentiate $$f(x) = (3x^2$ + 2)^5$

Question 16

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

Accepted Answer

$$\frac{1}{x}$$

Question 17

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

Accepted Answer

$$-3$$

Question 18

Which of the following is NOT a valid interpretation of the derivative?

Accepted Answer

The area under the curve.

Question 19

If $$y = x^x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 20

Suppose $$f(x) is $$not differentiable at $$x = 2. $$Which of the following could be true?

Accepted Answer

The graph of $$f$$ has a sharp corner at $$x = 2.$$

Calculus Study Guide: Derivatives and Their Applications

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