📚 Topic Summary
The Generalized Power Rule is a powerful tool in calculus that allows us to differentiate composite functions where one function is raised to a power. In simpler terms, if you have a function inside another function, and that entire inner function is raised to some power, the Generalized Power Rule tells you how to find the derivative. It's an extension of the regular Power Rule and uses the chain rule.
The rule states: If $y = [f(x)]^n$, then $\frac{dy}{dx} = n[f(x)]^{n-1} \cdot f'(x)$. Remember to always multiply by the derivative of the inner function, $f'(x)$. This is the key to applying the Generalized Power Rule correctly!
🧠 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Derivative |
a. A function composed of another function. |
| 2. Chain Rule |
b. The process of finding the derivative. |
| 3. Composite Function |
c. The instantaneous rate of change of a function. |
| 4. Power Rule |
d. A rule for differentiating a power of x. |
| 5. Differentiation |
e. A formula for differentiating composite functions. |
✍️ Part B: Fill in the Blanks
The Generalized Power Rule is an extension of the standard ______ Rule. It is used to find the derivative of a ______ function raised to a power. The key step is to multiply by the ______ of the inner function after applying the power rule to the ______ function.
🤔 Part C: Critical Thinking
Explain, in your own words, why it's important to use the chain rule in conjunction with the power rule when applying the Generalized Power Rule. Give an example of a function where using only the Power Rule would lead to an incorrect derivative. Show both correct and incorrect solutions.