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:
- <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>