What is the product rule and how does it work?

The product rule is a mathematical rule used to find the derivative of a function that is the product of two other functions. It is often used in calculus.

The product rule states that if we have two functions f(x) and g(x), then the derivative of their product f(x)g(x) is given by:

(f(x)g(x))’ = f'(x)g(x) + f(x)g'(x)

In other words, to find the derivative of the product of two functions, we take the derivative of the first function multiplied by the second function, and then add it to the derivative of the second function multiplied by the first function.

Let’s consider an example to understand how the product rule works. Suppose we have two functions f(x) = x^2 and g(x) = e^x. The derivative of f(x) is f'(x) = 2x, and the derivative of g(x) is g'(x) = e^x. 

By applying the product rule, we can find the derivative of the product f(x)g(x):

(f(x)g(x))’ = (x^2)(e^x)’ + (x^2)'(e^x)

            = 2x(e^x) + (2)(x)(e^x)

            = 2xe^x + 2xe^x

            = 4xe^x

So, the derivative of f(x)g(x) is 4xe^x.

The product rule can be a helpful tool in finding the derivatives of more complex functions that can be expressed as products of simpler functions. It allows us to break down the derivative calculation into smaller, more manageable parts.

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