The nested chain rule is used when differentiating composite functions, where one function is inside another. Essentially, you differentiate the outermost function, keeping the inner function intact, and then multiply by the derivative of the inner function. This process is repeated for each 'layer' of the composite function. The key is to identify the 'layers' and apply the chain rule iteratively. Visualizing the function as a series of nested boxes can be helpful. For example, if you have $f(g(h(x)))$, you would first find $f'(g(h(x)))$, then multiply by $g'(h(x))$, and finally multiply by $h'(x)$. This gives you $f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)$. Practice identifying these layers and applying the chain rule step-by-step.
Match the term with its definition:
| Term | Definition |
|---|---|
| 1. Composite Function | A. The derivative of the outer function multiplied by the derivative of the inner function. |
| 2. Chain Rule | B. A function formed by substituting one function into another. |
| 3. Derivative | C. A function that represents the instantaneous rate of change of another function. |
| 4. Inner Function | D. The function inside another function in a composite function. |
| 5. Outer Function | E. The function that contains another function in a composite function. |
Answers:
The nested chain rule is applied to __________ functions. You differentiate the __________ function first, keeping the __________ function intact. Then, you multiply by the __________ of the inner function. This process is repeated for each __________ of the function.
Word Bank: (layers, composite, outer, inner, derivative)
Explain, in your own words, how the chain rule can be applied multiple times when dealing with nested functions. Provide an example to illustrate your explanation.
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