Product Rule - Proof of The Product Rule

Proof of The Product Rule

A rigorous proof of the product rule can be given using the properties of limits and the definition of the derivative as a limit of Newton's difference quotient.

If

and ƒ and g are each differentiable at the fixed number x, then

Now the difference

is the area of the big rectangle minus the area of the small rectangle in the illustration.

The region between the smaller and larger rectangle can be split into two rectangles, the sum of whose areas is

Therefore the expression in (1) is equal to

Assuming that all limits used exist, (4) is equal to

\left(\lim_{w\to x}f(x)\right) \left(\lim_{w\to x} {g(w) - g(x) \over w - x}\right) + \left(\lim_{w\to x} g(w)\right) \left(\lim_{w\to x} {f(w) - f(x) \over w - x} \right). \qquad\qquad(5)

Now

This holds because f(x) remains constant as w → x.

This holds because differentiable functions are continuous (g is assumed differentiable in the statement of the product rule).

Also:

and

because f and g are differentiable at x;

We conclude that the expression in (5) is equal to

Read more about this topic:  Product Rule

Famous quotes containing the words proof of the, proof of, proof, product and/or rule:

    “Sculpture and painting are very justly called liberal arts; a lively and strong imagination, together with a just observation, being absolutely necessary to excel in either; which, in my opinion, is by no means the case of music, though called a liberal art, and now in Italy placed even above the other two—a proof of the decline of that country.”
    —Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

    “The thing with Catholicism, the same as all religions, is that it teaches what should be, which seems rather incorrect. This is “what should be.” Now, if you’re taught to live up to a “what should be” that never existed—only an occult superstition, no proof of this “should be”Mthen you can sit on a jury and indict easily, you can cast the first stone, you can burn Adolf Eichmann, like that!”
    —Lenny Bruce (1925–1966)

    “O, popular applause! what heart of man
    Is proof against thy sweet, seducing charms?”
    —William Cowper (1731–1800)

    “In fast-moving, progress-conscious America, the consumer expects to be dizzied by progress. If he could completely understand advertising jargon he would be badly disappointed. The half-intelligibility which we expect, or even hope, to find in the latest product language personally reassures each of us that progress is being made: that the pace exceeds our ability to follow.”
    —Daniel J. Boorstin (b. 1914)

    “All you people don’t know about lost causes. Mr. Paine does. He said once they were the only causes worth fighting for, and he fought for them once, for the only reason that any man ever fights for them. Because of just one plain, simple rule—Love Thy Neighbor. And in this world today, full of hatred, a man who knows that one rule has a great trust.”
    —Sidney Buchman (1902–1975)