Introduction: If $y(x)$ is the product of two functions $u(x)$ and $v(x)$, then we can apply the product rule to determine the derivative of $y(x)$.
Rule: Let $y(x) = u(x) \cdot v(x)$. Then:
$$ y'(x) = u'(x)v(x) + u(x)v'(x).$$
Example: Consider the function $y(x) = 5x^2 \cdot \ln(x)$. This function can be denoted as $y(x)=u(x) \cdot v(x)$, with $u(x)=5x^2$ and $v(x) = \ln(x)$. The derivative of $y(x)$ can be determined as follows (one might consider Derivatives of elementary functions and the example at Sum rule):
$$\begin{align}
u'(x) &= 5 \cdot 2x = 10x,\\
v'(x) &= \dfrac{1}{x},\\
y'(x) &= u'(x)v(x) + u(x)v'(x) = 10x\ln(x) + 5x^2\dfrac{1}{x} = 10x\ln(x) + \dfrac{5x^2}{x} = 10x\ln(x) + 5x.
\end{align}$$
