Linearity of differentiation

Let f and g be functions. Now consider:

<math>{d \over dx}(af(x) + bg(x))</math>

By the sum rule in differentiation, this is:

<math>{d \over dx}(af(x)) + {d \over dx}(bg(x))</math>

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

<math>af\ '(x) + bg'(x)</math>

Hence we have:

<math>{d \over dx}(af(x) + bg(x)) = af\ '(x) + bg'(x)</math>

Omitting the brackets[?], this is often written as:

<math>(af + bg)' = af\ '+ bg'</math>