Calculus and Its Applications, Chapter 1, 1.6, Section 1.6, Problem 2

Take the derivative of $y = x^9 \cdot x^4$: first, use the Product Rule; then,
by multiplying the expression before differentiating. Compare your results as a check.
By using Product Rule,

$
\begin{equation}
\begin{aligned}
y' = \frac{d}{dx} \left[ x^9 \cdot x^4 \right] &= x^9 \cdot \frac{d}{dx} (x^4) + x^4 \cdot \frac{d}{dx} (x^9)\\
\\
&= x^9 (4x^3) + x^4 (9x^8)\\
\\
&= 4x^{12} + 9x^{12}\\
\\
&= 13x^{12}
\end{aligned}
\end{equation}
$


By multiplying the expression first,

$
\begin{equation}
\begin{aligned}
y &= x^9 \cdot x^4 = x^{9 + 4} = x^{13}\\
\\
y' &= \frac{d}{dx}(x^{13}) = 13x^{12}
\end{aligned}
\end{equation}
$


Both results agree.