### Theorem If $u$ is a differentiable function of $x$, and $c$ is a constant, then $ \frac{d}{d x}(c u)=c \frac{d u}{d x} . $ In particular, if $n$ is any real number, then $ \frac{d}{d x}\left(c x^{n}\right)=c n x^{n-1} . $ #### Proof $ \begin{aligned} \frac{d}{d x} c u &=\lim _{h \rightarrow 0} \frac{c u(x+h)-c u(x)}{h} & & \begin{array}{l} \text { Derivative definition } \\ \text { with } f(x)=c u(x) \end{array} \\ &=c \lim _{h \rightarrow 0} \frac{u(x+h)-u(x)}{h} & & \text { Constant Multiple Limit Property } \\ &=c \frac{d u}{d x} & & u \text { is differentiable. } \end{aligned} $