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 we can say that the act of differentiation is linear, or the differential operator is a linear operator.

Let f and g be functions, with $\alpha$ and $\beta$ 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:

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

Hence we have:

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

Omitting the brackets, this is often written as:

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