Product Rule
The product rule is a formula used in calculus to find the derivative of a product of two functions
The product rule is a formula used in calculus to find the derivative of a product of two functions. It states that if you have two functions, f(x) and g(x), and you want to find the derivative of their product, then the derivative is given by the formula:
(d/dx)(f(x) * g(x)) = f'(x) * g(x) + f(x) * g'(x)
In words, the product rule states that to find the derivative of a product of two functions, you take the derivative of the first function and multiply it by the second function, then add it to the derivative of the second function multiplied by the first function.
Let’s go through an example to illustrate how to use the product rule:
Example: Find the derivative of f(x) = x^2 * sin(x)
Solution: We have a product of two functions, f(x) = x^2 and g(x) = sin(x). To find the derivative, we’ll use the product rule.
First, we need to find the derivatives of f(x) and g(x). The derivative of f(x) = x^2 is f'(x) = 2x, and the derivative of g(x) = sin(x) is g'(x) = cos(x).
Then, we can apply the product rule formula:
(d/dx)(f(x) * g(x)) = f'(x) * g(x) + f(x) * g'(x)
= (2x) * (sin(x)) + (x^2) * (cos(x))
Simplifying further, we get:
= 2x * sin(x) + x^2 * cos(x)
So, the derivative of f(x) = x^2 * sin(x) is 2x * sin(x) + x^2 * cos(x).
Remember to always use the product rule when taking the derivative of a product of two functions.
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