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

[math]\displaystyle{ g(x) = k \cdot f(x). }[/math]

where k is a constant.

Use the formula for differentiation from first principles to obtain:

[math]\displaystyle{ \begin{align} g'(x) & = \lim_{h \to 0} \frac{g(x+h)-g(x)}{h} \\ & = \lim_{h \to 0} \frac{k \cdot f(x+h) - k \cdot f(x)}{h} \\ & = \lim_{h \to 0} \frac{k(f(x+h) - f(x))}{h} \\ & = k \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \\ & = k \cdot f'(x). \end{align} }[/math]

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

In Leibniz's notation, this reads

[math]\displaystyle{ \frac{d(k \cdot f(x))}{dx} = k \cdot \frac{d(f(x))}{dx}. }[/math]

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

[math]\displaystyle{ \frac{d(-y)}{dx} = -\frac{dy}{dx}. }[/math]