The product rule is one of the differentiation rules we can use for finding derivatives. It's useful when we have two functions being multiplied together.
For example, let's say we have a function $h$, which multiplies $f$ and $g$.
These functions are themselves polynomial expressions, defined as follows.
$h$ is therefore the product of $f$ and $g$, and we can express it as follows.
Ok, now what happens when we want to find the derivative of $h$? The product rule states that, in this case, the derivative of h will be the derivative of f times g, plus the derivative of g plus f.
We can then start by first differentiating our $f$ and $g$, mostly by using the power rule.
Once these are found, we can just swap in the functions and derivatives, and come up with the expression for the derivative of the parent function $h$.