AP Calculus AB Derivative Rules Test: Nine Questions, Printable PDF

AP Calculus Derivative Rules Test: questions 1–5

1. Limit definition (2 points)

Let \(f(x)=x^{2}-3x\). Use the limit definition of the derivative to find \(f^{\prime}(x)\).

2. Basic rules (3 points)

Find the derivative of each function.

1. \(f(x)=5x^{4}-2\sqrt{x}+7\) (1 point)
2. \(g(x)=3e^{x}-4\sin x+2\ln x\) (1 point)
3. \(h(x)=\frac{6}{x^{2}}+3\cos x\) (1 point)

3. Product rule (2 points)

Let \(f(x)=x^{3}e^{x}\). Find \(f^{\prime}(x)\) and evaluate \(f^{\prime}(1)\).

4. Quotient rule (2 points)

Let \(g(x)=\frac{2x+1}{x^{2}+1}\). Find \(g^{\prime}(x)\) and evaluate \(g^{\prime}(0)\).

5. Tangent lines to a cubic (3 points)

The graph of \(f(x)=x^{3}-3x\) is shown.

1. Write an equation of the tangent line at \(x=2\). (1 point)
2. Find every \(x\) where the tangent line is horizontal. (1 point)
3. Find every \(x\) where the tangent line is parallel to \(y=6x\). (1 point)

AP Calculus Derivative Rules Test: questions 6–9

6. Differentiable piecewise function (2 points)

Let \(f(x)=x^{2}+1\) when \(x\le 2\) and \(f(x)=ax+b\) when \(x > 2\). Find \(a\) and \(b\) so that \(f\) is differentiable at \(x=2\).

7. Estimating rates from data (2 points)

The temperature \(T\) in \(^{\circ}\)F of a cup of coffee is measured \(t\) minutes after it is poured.

1. Find the average rate of change of \(T\) on \([0,4]\). Include units. (1 point)
2. Estimate \(T^{\prime}(6)\) using the points at \(t=4\) and \(t=8\), and explain what it means. (1 point)

8. Derivative with tangent (2 points)

Let \(f(x)=x\tan x\). Find \(f^{\prime}(x)\) and evaluate \(f^{\prime}(\frac{\pi}{4})\).

9. Motion along a line (2 points)

A particle moves along a line with position \(s(t)=t^{3}-6t^{2}+9t\) feet, \(t\ge 0\) seconds.

1. Find the velocity \(v(t)=s^{\prime}(t)\). (1 point)
2. Find every time the particle is at rest, and the position at each. (1 point)

Where do the points concentrate on this test?

Two questions (2 and 5) are weighted at three points, and the rest are weighted at two. Therefore the basic rules and the tangent question deserve a careful pass. Question 1 asks for a proof-style limit, so write each algebra line.

How do you make a piecewise function differentiable?

In question 6, \(f(x)=x^2+1\) for \(x\le2\) and \(ax+b\) after that. Match the values at 2, so \(2a+b=5\). Then match the slopes, since \(2x\) equals 4 at \(x=2\), which gives \(a=4\) and \(b=-3\).

Which AP calculus derivative rules test question needs a quotient?

Question 4 differentiates \(g(x)=\frac{2x+1}{x^2+1}\). Apply low times derivative of high minus high times derivative of low, over low squared. At \(x=0\), the result is 2.

What do the tangent and motion questions ask?

Question 5 finds the tangent to \(x^3-3x\) at \(x=2\), where the slope is 9. Question 9 uses \(s(t)=t^3-6t^2+9t\), so the velocity \(3(t-1)(t-3)\) shows rest at 1 and 3 seconds.

What is the best review step after the derivative test?

Mark each lost point as definition, rule choice or algebra. Then redo the product and quotient questions with new functions. Finally, explain to a classmate why differentiability requires matching slopes as well as values.

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AP Calculus AB Definition of the Derivative and Basic Rules test

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