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# 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:

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

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