The product rule is a formula used in calculus to find the derivatives of products of two or more functions. It is used when we have a function f multiplied by another function g, and we want to find the derivative of the product fg. The product rule states that the derivative of fg is equal to the derivative of f times g plus f times the derivative of g. In mathematical notation, this can be written as:

$$(fg) = fg + fg$$

where f and g represent the derivatives of f and g, respectively.

To better understand the product rule, lets work through an example. Suppose we want to differentiate the function f(x) = x^2 sin(x). We can use the product rule to find its derivative as follows:

$$f(x) = (x^2) \sin(x) + x^2 (\sin(x))$$ $$f(x) = 2x \sin(x) + x^2 \cos(x)$$

Therefore, the derivative of f(x) is f(x) = 2x sin(x) + x^2 cos(x) .

It is important to note that the product rule only applies when we have a product of two or more functions. If we have a product of more than two factors, we can use the generalized product rule. Additionally, we should be careful not to confuse the product rule with the quotient rule, which is used to find the derivative of a quotient of two functions.