Constant multiple rule

In calculus, the constant factor rule in differentiation allows one to take constants outside a derivative and concentrate on differentiating the function of x itself. This is a part of the linearity of differentiation.

Consider a differentiable function

$g(x)=k\cdot f(x).$

where k is a constant.

Use the formula for differentiation from first principles to obtain:

$\begin{array}{rl}{g}^{\prime }(x)& =\underset{h\to 0}{\lim }\frac{g(x+h)-g(x)}{h}\\ & =\underset{h\to 0}{\lim }\frac{k\cdot f(x+h)-k\cdot f(x)}{h}\\ & =\underset{h\to 0}{\lim }\frac{k(f(x+h)-f(x))}{h}\\ & =k\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}\\ & =k\cdot f^{\prime }(x).\end{array}$

This is the statement of the constant factor rule in differentiation, in Lagrange's notation for differentiation.

In Leibniz's notation, this reads

$\frac{d(k\cdot f(x))}{dx}=k\cdot \frac{d(f(x))}{dx}.$

If we put k = −1 in the constant factor rule for differentiation, we have:

$\frac{d(-y)}{dx}=-\frac{dy}{dx}.$