AP Calculus AB: Derivatives Practice Questions
Test yourself on Derivatives with 10 original AP Exams practice questions. Pick an answer to see instant feedback and a full explanation.
Free original practice questions for study purposes. Open Exam Prep is an independent study resource and is not affiliated with, endorsed by, or sponsored by the makers of AP Exams.
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Answer the questions below — you get instant feedback and a full explanation for each.
1. For f(x)=3x^2-5x+4, what is f'(x)?
Explanation. Use the power rule term by term: the derivative of 3x^2 is 6x, the derivative of -5x is -5, and the derivative of 4 is 0.
2. If g(x)=x^4-2x^3+x, what is g'(1)?
Explanation. g'(x)=4x^3-6x^2+1. At x=1, g'(1)=4-6+1=-1.
3. A particle has position s(t)=t^3-6t^2+9t meters. What is its velocity at t=2?
Explanation. Velocity is the derivative of position. s'(t)=3t^2-12t+9, so s'(2)=12-24+9=-3 m/s.
4. For y=(2x+1)^5, which expression gives dy/dx?
Explanation. Apply the chain rule: derivative of u^5 is 5u^4, and u=2x+1 has derivative 2. The product is 10(2x+1)^4.
5. What is the slope of the tangent line to y=x^2+2x at x=3?
Explanation. The tangent slope is y'. Since y'=2x+2, substituting x=3 gives 2(3)+2=8.
6. If h(x)=sin x + x^2, what is h'(x)?
Explanation. Differentiate each term: the derivative of sin x is cos x, and the derivative of x^2 is 2x.
7. For f(x)=x^2 e^x, which expression is f'(x)?
Explanation. Use the product rule: (x^2)'e^x + x^2(e^x)' = 2xe^x + x^2e^x = e^x(x^2+2x).
8. At which x-value does f(x)=x^3-3x have a horizontal tangent?
Explanation. Horizontal tangents occur when f'(x)=0. Here f'(x)=3x^2-3=3(x^2-1), so x=1 or x=-1.
9. If y=ln(3x), for x>0, what is dy/dx?
Explanation. Using the chain rule, d/dx[ln(3x)] = (1/(3x)) * 3 = 1/x.
10. The function f is differentiable and f'(2)=7. What does this value represent?
Explanation. A derivative value gives the instantaneous rate of change, equivalently the slope of the tangent line, at that input.
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FAQ
What derivative rules matter most for AP Calculus AB?
Know the power, product, quotient, and chain rules, plus basic trig, exponential, and logarithmic derivatives.
How should I practice derivative questions?
Mix symbolic differentiation with interpretation questions about slope, velocity, tangent lines, and rates of change.
What is a common derivative mistake?
Many students forget the inner derivative in chain rule problems, especially with powers, trig functions, and logarithms.