Linearity of differentiation

In mathematics, the linearity of differentiation is a most fundamental property of the derivative, in differential calculus. It follows from the sum rule in differentiation and the constant factor rule in differentiation. Thus it can be said that the act of differentiation is linear, or the differential operator is a linear operator.

Let f and g be functions, with α and β fixed. Now consider:

$\frac{\mbox{d}}{\mbox{d} x} ( \alpha \cdot f(x) + \beta \cdot g(x) )$

By the sum rule in differentiation, this is:

$\frac{\mbox{d}}{\mbox{d} x} ( \alpha \cdot f(x) ) + \frac{\mbox{d}}{\mbox{d} x} (\beta \cdot g(x))$

By the constant factor rule in differentiation, this reduces to:

$\alpha \cdot f'(x) + \beta \cdot g'(x)$

This in turn leads to:

$\frac{\mbox{d}}{\mbox{d} x}(\alpha \cdot f(x) + \beta \cdot g(x)) = \alpha \cdot f'(x) + \beta \cdot g'(x)$

Omitting the brackets, this is often written as:

$(\alpha \cdot f + \beta \cdot g)' = \alpha \cdot f'+ \beta \cdot g'$