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:

    “The source of Pyrrhonism comes from failing to distinguish between a demonstration, a proof and a probability. A demonstration supposes that the contradictory idea is impossible; a proof of fact is where all the reasons lead to belief, without there being any pretext for doubt; a probability is where the reasons for belief are stronger than those for doubting.”
    —Andrew Michael Ramsay (1686–1743)

    “Ah! I have penetrated to those meadows on the morning of many a first spring day, jumping from hummock to hummock, from willow root to willow root, when the wild river valley and the woods were bathed in so pure and bright a light as would have waked the dead, if they had been slumbering in their graves, as some suppose. There needs no stronger proof of immortality. All things must live in such a light. O Death, where was thy sting? O Grave, where was thy victory, then?”
    —Henry David Thoreau (1817–1862)

    “The proof of a poet is that his country absorbs him as affectionately as he has absorbed it.”
    —Walt Whitman (1819–1892)