The product rule is a fundamental concept in calculus that helps us find the derivative of a function that is the product of two or more functions. It states that the derivative of $u(x)v(x)$ with respect to $x$ is $u'(x)v(x) + u(x)v'(x)$. In simpler terms, you take the derivative of the first function, multiply it by the second function, then add that to the first function multiplied by the derivative of the second function. Mastering this rule is essential for more advanced calculus topics! This worksheet gives you a chance to practice. Let's go!
Match the term with its correct definition:
| Term | Definition |
|---|---|
| 1. Derivative | A. A function representing the instantaneous rate of change of another function. |
| 2. Product Rule | B. The process of finding the derivative of a function. |
| 3. Differentiation | C. A rule that finds the derivative of a function that is the product of two or more functions. |
| 4. Function | D. An expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). |
| 5. Instantaneous Rate of Change | E. The rate of change of a function at a specific point. |
Match the correct Term with the corresponding definition: 1-A, 2-C, 3-B, 4-D, 5-E
The product rule states that if we have a function $h(x)$ that is the product of two functions, $f(x)$ and $g(x)$, such that $h(x) = f(x) \cdot g(x)$, then the derivative of $h(x)$, denoted as $h'(x)$, is given by $h'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)$. This means we take the _____ of the first function and multiply it by the _____ function, then add the first function multiplied by the _____ of the second function. Therefore, to use the product rule you need to know how to take the _____ of simple polynomial functions.
Answers: derivative, second, derivative, derivative
Explain in your own words why the product rule is necessary. Can you provide an example of a situation where you couldn't find the derivative without it?
Answer: The product rule is necessary because the derivative of a product of functions is *not* simply the product of the derivatives. The product rule accounts for how each function in the product influences the overall rate of change. For example, consider $h(x) = x \cdot sin(x)$. We cannot simply take the derivative of $x$ and $sin(x)$ separately and multiply them to get the correct derivative of $h(x)$. The product rule allows us to correctly find $h'(x) = sin(x) + x \cdot cos(x)$. Without the product rule, differentiating this would be very complicated.
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