AP (Advanced Placement) · AP Calculus BC

Rates of change and the definition of the derivative: Practice Questions

5 multiple-choice questions marked as you go, and 5 written questions with worked solutions. All on Rates of change and the definition of the derivative.

10 questions17 marksFree, no account

Question 1

1 mark

Evaluate the derivative of \( f(x) = \frac{2x - 3}{x^2 + 1} \) at \( x = 1 \).

Question 2

1 mark

Using the limit definition of the derivative, find \(f'(x)\) for \(f(x) = \sqrt{3x + 1}\).

Question 3

1 mark

Find the \(n\)-th derivative of \(f(x) = x e^x\).

Question 4

1 mark

Find the derivative of the function \( f(x) = 2x^3 - e^x + \ln(4) \) with respect to \( x \).

Question 5

1 mark

Find the derivative of \(f(x) = x^3 e^{2x}\) with respect to \(x\).

Question 6

2 marks

Calculate the derivative of \( f(x) = 4e^x - \frac{2}{x^3} \) with respect to \( x \).

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Question 7

2 marks

Using the basic power rule and constant multiple rule, calculate the derivative of the function \( f(x) = \frac{4}{\sqrt{x}} + 5\pi \) with respect to \( x \).

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Question 8

2 marks

Using the product rule, find the derivative of \( y = x^2 \sin(x) \).

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Question 9

3 marks

Consider the function defined by \( f(x) = \frac{1}{x+2} \).
(a) Using the formal definition of the derivative as a limit, find \( f'(x) \).
(b) Evaluate \( f'(1) \).

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Question 10

3 marks

Consider the function \( f(x) = x^4 - 2x + 5 \).
(a) Using the fundamental power rule and linearity of the derivative, find \( f'(x) \).
(b) Find the value of \( x \) such that the instantaneous rate of change of \( f(x) \) is 2.
(c) Determine the coordinates of the point on the graph where the tangent line is horizontal.

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