Sum Rule in Differentiation - Proof

Proof

Let y be a function given by the sum of two functions u and v, such that:

Now let y, u and v be increased by small increases Δy, Δu and Δv respectively. Hence:

So:

Now divide throughout by Δx:

Let Δx tend to 0:

Now recall that y = u + v, giving the sum rule in differentiation:

The rule can be extended to subtraction, as follows:

Now use the special case of the constant factor rule in differentiation with k=−1 to obtain:

Therefore, the sum rule can be extended so it "accepts" addition and subtraction as follows:

The sum rule in differentiation can be used as part of the derivation for both the sum rule in integration and linearity of differentiation.

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