Linearity of differentiation

A property of the derivative which follows from the sum rule in differentiation and the constant factor rule in differentiation.

Let f and g be functions. Now consider:

d/dx(af(x) + bg(x))

By the sum rule in differentiation, this is:

d/dx(af(x)) + d/dx(bg'(x))

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

af'(x) + bg'(x)

Hence we have:

d/dx(af(x) + bg(x)) = af'(x) + bg'(x)

Omitting the brackets, this is often written as:

(af + bg)' = af' + bg'