The Product Rule

Consider $f(x)=x^2$ and $g(x)=x^5$. Applying the power rule, we get $f'(x)=2x$ and $g'(x)=5x^4$. Now, $f(x)\cdot g(x)=x^2\cdot x^5=x^7$, so we have $\big(f(x)\cdot g(x)\big)'=7x^6$, again by the power rule. However, $$f'(x)\cdot g'(x)=2x\cdot 5x^4=10x^5\ne 7x^6!$$ In particular, this shows that $\big(f(x)g(x)\big)'\ne f'(x)\cdot g'(x)$. Instead,

The Product Rule: $$ \frac{d}{dx}\big(f(x)\cdot g(x)\big) = f'(x)\cdot g(x) + f(x)\cdot g'(x).$$

Think about what would happen if $f(x)$ was a constant and $g(x)$ was changing. Then, by the constant multiple rule, the derivative of $f(x)\cdot g(x)$ would be $f(x)\cdot g'(x)$. Likewise, if $f(x)$ was changing and $g(x)$ was a constant, then we would get $f'(x)\cdot g(x)$. When $f(x)$ and $g(x)$ are both changing, the actual derivative of $f(x)\cdot g(x)$ is the sum of two terms, one proportional to how much $f(x)$ is changing and the other proportional to how much $g(x)$ is changing.

Example: Since the derivative of $x^2$ is $2x$ and the derivative of $e^x$ is $e^x$, the derivative of $x^2e^x$ is $$ (x^2 e^x)' = (x^2)' e^x + x^2 (e^x)' = 2x e^x + x^2 e^x = (x^2+2x)e^x.$$