Recognizing a Composite Function
A composite function is a function built by plugging one function into another. In function notation, it looks like : the function is applied to the output of .
For example, consider . You can think of this as two separate operations:
- First, calculate .
- Then raise that result to the fifth power.
That is exactly the kind of nested structure the chain rule is designed to handle.
The Chain Rule
If , then the chain rule says:
The key idea is simple: differentiate the outside, keep the inside intact, and then multiply by the derivative of the inside.
How to Apply the Chain Rule
- Find the inside function. Look for the expression being operated on by another function.
- Identify the outside function. Ask what operation is being performed on that entire inside expression.
- Differentiate the outside function while treating the inside expression as a single variable.
- Multiply by the derivative of the inside function.
- Simplify your answer only after both derivatives have been taken.
Worked Examples
Differentiate .
Differentiate .
Differentiate .
Differentiate .
Differentiate . This function requires the chain rule more than once.
Why the Chain Rule Works
The chain rule reflects the idea that a small change in the input travels through several functions. If changes with respect to , and changes with respect to , then the total rate of change is found by multiplying those rates together.
This is why the chain rule is especially important in applications. It lets you connect rates of change through multiple stages of a process.
Common Mistakes
A common incorrect answer to is . The missing factor is the derivative of the inside function, .
When differentiating , first treat as a single unit. The outside derivative is . Only after that do you multiply by the derivative of the inside.
Some functions contain multiple nested functions. For , the chain rule must be applied through the outer, middle, and inner layers. Look for every nested function before declaring the derivative finished.
The power rule alone works for . But when the base is itself a function, such as , the chain rule supplies the additional factor.
Practice Problems
Differentiate .
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Differentiate .
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Differentiate .
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Differentiate .
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