Linearity of differentiation

Linearity of differentiation is 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:

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

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:

$(af+bg)'=af''+b(g)$