What This Practice Set Covers
Differentiation is one of the central skills in AP Calculus. Before working with the chain rule, implicit differentiation, related rates, or optimization, you should be comfortable finding derivatives quickly and accurately.
This practice set reviews the essential basics: interpreting a derivative, using the limit definition, applying the power rule, differentiating sums and differences, working with constants, and interpreting derivatives in context.
Essential Derivative Rules
Keep these rules available as you work through the problems. The goal is not simply to memorize formulas, but to recognize which rule applies immediately.
How to Approach Each Problem
- Identify what is being asked. Is the problem asking for a derivative function, a derivative at a point, a tangent line, or an interpretation?
- Look at the structure of the function. A polynomial may require only the power rule, while a product, quotient, or composition may require another rule.
- Differentiate carefully. Keep constants, exponents, and signs under control.
- Check your answer. Make sure the derivative has the expected behavior and that you evaluated at the requested point when necessary.
Worked Examples
Find the derivative of .
Suppose . Find .
Use the limit definition to find the derivative of .
Find the equation of the tangent line to at .
Common Mistakes
When using the power rule, multiply by the original exponent and then reduce the exponent by 1. For example, , not .
The derivative of any constant is zero. A term such as contributes nothing to the derivative.
The value tells you the height of the function at a point. The value tells you the instantaneous rate of change there. They are generally different numbers.
If a problem asks for , finding is only the first step. You still need to substitute the specified value of .
Practice Problems
Differentiate .
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Find if .
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Differentiate .
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If , find .
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Use the limit definition of the derivative to find the derivative of .
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Find the equation of the tangent line to at .
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The position of a particle is meters at time seconds. Find the particle's instantaneous velocity at .
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Find the derivative of .
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Quick Self-Check
Before moving on, you should be able to answer these questions without hesitation:
- What does represent geometrically?
- What is the derivative of a constant?
- What happens to the exponent when you apply the power rule?
- How do you find the slope of a tangent line at ?
- What is the difference between average and instantaneous rate of change?
- When should you use the limit definition rather than a memorized derivative rule?
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