📚 Topic Summary
The product rule is a fundamental concept in calculus used to find the derivative of a function that is the product of two or more functions. Simply put, if you have a function $f(x) = u(x)v(x)$, then the derivative $f'(x)$ is given by $f'(x) = u'(x)v(x) + u(x)v'(x)$. When dealing with *advanced* product rule problems, the challenge often lies in applying the rule multiple times, dealing with complex functions for $u(x)$ and $v(x)$, or combining the product rule with other differentiation techniques like the chain rule.
Advanced problems might also involve trigonometric functions, exponential functions, or logarithmic functions within the product. Careful attention to algebraic manipulation and simplification is key to arriving at the correct answer.
🔤 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Derivative |
A. The rate at which a function changes with respect to its variable. |
| 2. Product Rule |
B. A method used to differentiate the product of two or more functions. |
| 3. Chain Rule |
C. A method used to differentiate composite functions. |
| 4. Function |
D. A relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. |
| 5. Differentiation |
E. The process of finding the derivative of a function. |
Match the numbers 1-5 with the letters A-E.
✍️ Part B: Fill in the Blanks
Complete the following paragraph using the words provided:
The _________ rule is used when we need to find the derivative of a product of two or more _________. The formula is $ (uv)' = u'v + _________. $ When applying this rule to more complex problems, such as those involving trigonometric or exponential functions, it's crucial to also consider the _________ rule and other differentiation _________. Careful _________ and simplification are essential.
Word Bank: Product, Functions, vu', Chain, Techniques, Algebra
🤔 Part C: Critical Thinking
Explain, in your own words, how the product rule is applied when you have a function that is the product of *three* functions, $f(x) = u(x)v(x)w(x)$. Provide an example.