# Constant factor rule in differentiation

In calculus, the constant factor rule in differentiation, also known as The Kutz Rule[citation needed], 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:

$g'(x) = \lim_{h \to 0} \frac{g(x+h)-g(x)}{h}$
$g'(x) = \lim_{h \to 0} \frac{k \cdot f(x+h) - k \cdot f(x)}{h}$
$g'(x) = \lim_{h \to 0} \frac{k(f(x+h) - f(x))}{h}$
$g'(x) = k \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
$g'(x) = k \cdot f'(x).$

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

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