Why Trigonometric Functions Make Implicit Differentiation Interesting
In ordinary differentiation, you usually start with an equation such as and differentiate directly with respect to . But in implicit differentiation, may appear throughout the equation, and you do not necessarily solve for it first.
Trigonometric functions add one more layer. If you see , you know the derivative of sine is cosine. But if you see , the argument of the sine is , not simply . The chain rule therefore appears automatically.
And if the trigonometric function contains , there is an additional factor from implicit differentiation. For example:
That is the feature that distinguishes implicit differentiation from ordinary chain-rule problems.
The Fundamental Idea
When differentiating an equation implicitly with respect to , treat as a function of . Therefore, whenever you differentiate a term involving , its derivative will generally contain .
Compare that with ordinary differentiation:
The difference is crucial. The extra appears because depends on .
Trigonometric Derivatives You Need
Before working through implicit examples, make sure you know the basic trigonometric derivatives:
In an implicit problem, each of these rules may also require the chain rule. For example:
The procedure is always the same: differentiate the trigonometric function first, then multiply by the derivative of its entire argument.
A Step-by-Step Method
- Differentiate both sides of the equation with respect to .
- Apply the ordinary trigonometric derivative rule. For example, sine becomes cosine and cosine becomes negative sine.
- Apply the chain rule to the argument. If the argument contains , its derivative will usually contain .
- Collect all terms containing .
- Factor out .
- Solve for .
Worked Examples
Find for the equation: .
Find for: .
Find if: .
Find if: .
Find if: .
What If the Trigonometric Function Contains Both x and y?
This is one of the most important patterns to recognize. Suppose you have: .
The entire expression is the input to the sine function. Therefore, the chain rule says:
Notice what happens inside the parentheses. The derivative of is , while the derivative of is .
This is exactly why you should avoid treating as though it were a constant. In implicit differentiation, is a function of .
Common Mistakes
If you differentiate as simply , you have treated as though it were independent of . The correct derivative is .
For , the derivative is not just . You must also differentiate the inside expression, producing .
Remember: . A very common error is writing instead of .
First differentiate the entire equation. Then collect every term containing on one side. Only after that should you factor and solve for .
A Reliable AP Calculus Strategy
When an implicit differentiation problem contains trigonometric functions, use this mental checklist:
- Differentiate everything. Do not try to isolate first.
- Whenever you differentiate something involving y, look for .
- For trigonometric compositions, apply the chain rule completely.
- Put every term together.
- Factor out .
- Divide to isolate .
On an AP free-response question, it is usually better to keep your algebra organized than to rush toward a simplified final expression. Correctly identifying where the terms come from is the most important part.
Practice Problems
Find if .
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Find if .
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Find if .
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Find if .
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