The first-principles approach to finding a derivative is time consuming, and often messy or difficult.

As you continue to solve derivatives, you'll find yourself doing the same manipulations over and over again for different problems.

Instead of wasting time like this, we can find efficient methods to differentiate (find the derivative of) new functions using differentiations that we already know.

For example, if we have a function $h(x) ={f(x)}{g(x)}$ and we want to find

then we can find a formula for this in terms of $f(x)$ and $g(x)$, since $h(x)$ is defined using them.

That is what we call the Product Rule.