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:

    “The fact that several men were able to become infatuated with that latrine is truly the proof of the decline of the men of this century.”
    —Charles Baudelaire (1821–1867)

    “There are some persons in this world, who, unable to give better proof of being wise, take a strange delight in showing what they think they have sagaciously read in mankind by uncharitable suspicions of them.”
    —Herman Melville (1819–1891)

    “From whichever angle one looks at it, the application of racial theories remains a striking proof of the lowered demands of public opinion upon the purity of critical judgment.”
    —Johan Huizinga (1872–1945)

    “Cultural expectations shade and color the images that parents- to-be form. The baby product ads, showing a woman serenely holding her child, looking blissfully and mysteriously contented, or the television parents, wisely and humorously solving problems, influence parents-to-be.”
    —Ellen Galinsky (20th century)

    “For all of us Frenchmen, the guiding rule of our epoch is to be faithful to France.”
    —Charles De Gaulle (1890–1970)