Sum Rule in Differentiation - Generalization To Sums

Generalization To Sums

Consider a set of functions f₁, f₂,..., f_n. Then

so

In other words, the derivative of any sum of functions is the sum of the derivatives of those functions.

This follows easily by induction; we have just proven this to be true for n = 2. Assume it is true for all n < k, then define

Then

and it follows from the proof above that

By the inductive hypothesis,

so

which ends the proof of the sum rule of differentiation.