Scalar product rule

If a function $y(x)$ is the product of a number $c$ (also called scalar $c$) and another function $u(x)$, then we can use the scalar product rule to determine the derivative of $y(x)$.

Rule: Let $y(x) = cu(x)$. Then:
$$y'(x) = c u'(x).$$

Example: Take the function $y(x) = 5x^2$. This function can be denoted as $y(x)=cu(x)$, with $c=5$ and $u(x)=x^2$. The derivative of $y(x)$ can be determined in two steps (one might consider Derivatives of elementary functions):
$$\begin{align} u'(x) &= 2x,\\ y'(x) &= 5 \cdot 2x = 10x. \end{align}$$