The Constant Multiple Rule for Derivatives

Theorem:

$$\frac{d}{dx} \left[ c \, f\,(x) \right] = c \, f\,'(x)$$

Proof:

$$\begin{array}{rcl} \displaystyle{\frac{d}{dx} \left[ c \, f\,(x) \right]} & = & \displaystyle{\lim_{h \rightarrow 0} \ \frac{c \, f\,(x+h) - c \, f\,(x)}{h}}\\\\ & = & \displaystyle{\lim_{h \rightarrow 0} \ c \cdot \frac{f\,(x+h) - f\,(x)}{h}}\\\\ & = & \displaystyle{c \cdot \lim_{h \rightarrow 0} \ \frac{f\,(x+h) - f\,(x)}{h}}\\\\ & = & \displaystyle{c \, f\,'(x)} \end{array}$$

QED.