The Product Rule is a formula used in calculus to find the derivative of the product of two functions.

What is the Product Rule?

If you have two functions, $f(x)$ and $g(x)$, and you want to differentiate their product $h(x) = f(x) \cdot g(x)$, the Product Rule states:

$\frac{d}{dx} \big[f(x) \cdot g(x)\big] = f'(x) \cdot g(x) + f(x) \cdot g'(x)$

This means:

1. Take the derivative of the first function ($f'(x)$) and multiply it by the second function ($g(x)$).
2. Add the first function ($f(x)$) multiplied by the derivative of the second function ($g'(x)$).

---

Why does it work?

The derivative measures how things change. When two functions are multiplied, their rates of change interact. The Product Rule ensures that both functions' contributions to the overall rate of change are accounted for.

---

Simple Example:

Let $f(x) = x^2$ and $g(x) = \sin(x)$. To differentiate $h(x) = f(x) \cdot g(x) = x^2 \cdot \sin(x)$:

1. Derivative of $f(x) = x^2$ is $f'(x) = 2x$.
2. Derivative of ( g(x) = \sin(x