📚 Topic Summary
When finding the derivative of a function that involves both a quotient and a composite function, you need to apply both the quotient rule and the chain rule. The quotient rule helps you differentiate functions that are fractions, while the chain rule helps you differentiate composite functions (functions inside other functions). Combining these rules involves carefully identifying the 'outer' and 'inner' functions and applying the rules in the correct order. Remember to work from the outside in!
🧮 Part A: Vocabulary
Match the term to its definition:
- Term: Quotient Rule
- Term: Chain Rule
- Term: Derivative
- Term: Composite Function
- Term: Function
- Definition: A function that results from the composition of two or more functions.
- Definition: A rule for differentiating the ratio of two functions.
- Definition: A rule for differentiating composite functions.
- Definition: A relation where each input has only one output.
- Definition: The instantaneous rate of change of a function.
✍️ Part B: Fill in the Blanks
The _______ rule is used to differentiate functions that are fractions. The _______ rule is used to differentiate composite functions. When applying the chain rule, you multiply the derivative of the _______ function by the derivative of the _______ function, evaluated at the inner function. Remember to apply these rules in the correct _______ to get the right answer!
🤔 Part C: Critical Thinking
Explain in your own words the strategy you would use to identify the outer and inner functions when asked to find the derivative of: $y = \frac{\sin(x^2)}{x}$