The product rule is a fundamental concept in calculus that allows us to find the derivative of a function that is the product of two or more functions. If we have a function $y = u(x)v(x)$, where $u(x)$ and $v(x)$ are differentiable functions, then the derivative of $y$ with respect to $x$ is given by the formula: $\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)$. In simpler terms, it's the derivative of the first function times the second function, plus the first function times the derivative of the second function. This rule is essential for differentiating complex functions and forms the basis for many other differentiation techniques.
Match the term with the correct definition:
| Term | Definition |
|---|---|
| 1. Product Rule | A. The rate at which a function's output changes with respect to its input. |
| 2. Derivative | B. A function that represents the slope of another function. |
| 3. Differentiable | C. The rule used to find the derivative of a function that is the product of two other functions. |
| 4. Function | D. Capable of having a derivative. |
| 5. Slope | E. 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. |
Match the numbers (1-5) to the letters (A-E).
Complete the following paragraph by filling in the missing words.
The ________ Rule is used when differentiating the ________ of two functions. If $y = u(x)v(x)$, then $\frac{dy}{dx}$ = ________ + $u(x)v'(x)$. The derivative of the ________ function is multiplied by the ________ function, and this is added to the first function multiplied by the ________ of the second function.
Explain in your own words why the product rule is necessary and provide an example of a function where it must be used.
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