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 $\big(f(x)g(x)\big)'\ne
f'(x)\cdot g'(x)$. We differentiate a product by the product
rule.

The derivative of a product is not the product of the
derivatives.

We write, briefly, that $(fg)'=f'g+fg'$ -- take the derivative of
one function and leave the other alone, then add to that the
derivative of the other function and leave the first alone.

We can extend the Product Rule to the product of three
functions: $(fgh)'=f'gh+fg'h+fgh'$.

DO: What do you think
$(fghp)'$ is?

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
$$ (x^2 e^x)' = \frac{d}{dx}(x^2) e^x + x^2 \frac{d}{dx}(e^x) = 2x
e^x + x^2 e^x = (x^2+2x)e^x.$$