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.

Read more about this topic:  Sum Rule In Differentiation

Famous quotes containing the word proof:

    “Right and proof are two crutches for everything bent and crooked that limps along.”
    —Franz Grillparzer (1791–1872)

    “Talk shows are proof that conversation is dead.”
    —Mason Cooley (b. 1927)

    “If some books are deemed most baneful and their sale forbid, how, then, with deadlier facts, not dreams of doting men? Those whom books will hurt will not be proof against events. Events, not books, should be forbid.”
    —Herman Melville (1819–1891)