How to Use This Practice Set
The chain rule and implicit differentiation are closely connected. The chain rule handles nested functions, while implicit differentiation applies derivative rules when y is not isolated. In both cases, the key skill is recognizing the structure of the expression before differentiating.
Try each problem without opening the answer. Write your derivative one step at a time, then use the expandable solution to check both your final answer and your reasoning.
For a composite function:
For implicit differentiation, remember that when differentiating a term involving , you must treat as a function of .
Part I: Chain Rule Practice
These first problems focus on recognizing an outside function and an inside function. The expressions become progressively more complicated.
Differentiate .
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Let . Then .
Differentiate .
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The outside function is sine and the inside function is .
Differentiate .
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The derivative of is itself, multiplied by the derivative of the exponent.
Differentiate .
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There are two layers of composition. First differentiate the outer square root, then multiply by the derivative of the inner expression.
Differentiate .
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Use the product rule first. The derivative of the sine term requires the chain rule.
Differentiate .
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Rewrite the denominator using a negative exponent before differentiating.
Part II: Implicit Differentiation Practice
In implicit differentiation, do not try to solve for first unless the problem specifically asks you to. Differentiate both sides with respect to , remembering that is a function of .
Find for the curve .
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Differentiate both sides:
Collect the terms containing and solve:
Find for .
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The product requires the product rule.
Solving for the derivative gives:
Find for .
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The term involving requires the chain rule.
Therefore:
Find for .
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Differentiate each side with respect to :
Now isolate the derivative:
Part III: Mixed AP Calculus Problems
These problems combine several rules. On the AP exam, you will often need to recognize the correct differentiation method rather than being told which rule to use.
Suppose . Find .
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The first term uses the chain rule and the second term also uses the chain rule.
Find the equation of the tangent line to at the point .
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Differentiate implicitly:
At the given point, the slope is:
Using point-slope form, the tangent line is:
Check Your Progress
For a 12-question set, a useful benchmark is: 10–12 correct means you are ready to move into harder mixed differentiation problems; 7–9 means review the rules you missed; 6 or fewer means revisit the chain rule and basic implicit differentiation before moving on.
Common Mistakes to Watch For
For , the derivative is not simply . You must multiply by the derivative of the inside function.
In implicit differentiation, is a function of . Therefore, differentiating produces an additional factor of .
After implicit differentiation, your equation may still contain . Collect all derivative terms on one side and factor before dividing to isolate .
A function can require several rules at once. Look for the largest structural feature first: product, quotient, or composition. Then apply the chain rule wherever a nested function appears.
A Reliable AP Exam Strategy
- Look at the structure before differentiating. Identify products, quotients, compositions, and implicit relationships.
- Keep nested expressions intact. Do not expand an expression simply because you can.
- Write the chain-rule factor explicitly. This prevents the most common differentiation error.
- For implicit equations, mark every derivative of y. Terms such as should immediately signal that you need to collect and solve for .
- Only simplify after the derivative is structurally correct. On an AP free-response question, an unsimplified but correct derivative can often be easier to verify than an aggressively simplified expression.
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