Linearity of differentiation

Differentiation is a linear operator; this 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:

${d \over dx}(af(x)+bg(x))$

By the sum rule in differentiation, this is:

${d \over dx}(af(x))+{d \over dx}(bg(x))$

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

$af\ '(x)+bg'(x)$

Hence we have:

${d \over dx}(af(x)+bg(x))=af\ '(x)+bg'(x)$

Omitting the brackets, this is often written as:

$(af+bg)'=af\ '+bg'$