Product Rule Discovery By Leibniz Examples Common Error

Discovery of this rule is credited to Leibniz, who demonstrated it using differentials. Here is Leibniz's argument: Let u(x) and v(x) be two differentiable functions of x. Then the differential of uv is

Since the term (du)(dv) is "negligible" (i.e. at least quadratic in du and dv), Leibniz concluded that

and this is indeed the differential form of the product rule. If we divide through by the differential dx, we obtain

2 Examples

Suppose you want to differentiate f(x) = x² sin(x). By using the product rule, you get the derivative f'(x) = 2x sin(x) + x²cos(x) (since the derivative of x² is 2x and the derivative of sin(x) is cos(x)).
One special case of the product rule is the Constant Multiple Rule which states: if c is a real number and f(x) is a differentiable function, then cf(x) is also differentiable, and its derivative is (c × f)'(x) = c × f'(x). (This follows from the product rule since the derivative of any constant is zero.) This combined with the sum rule for derivatives shows that differentiation is linear.
The product rule can be used to derive the rule for integration by parts and the quotient rule.

3 Common error

It is a common error, when studying calculus, to suppose that the derivative of (uv) equals (u′)(v′) (Leibniz himself made this error initially); however, it is quite easy to find counterexamples to this. Most simply, take a function f, whose derivative is f '(x). Now that function can also be written as f(x) · 1, since 1 is the identity element for multiplication. Suppose the above-mentioned misconception were true; if so, (u′)(v′) would equal zero; since, the derivative of a constantIn calculus, the derivative of a constant function is zero. A constant function is one that does not depend on the independent variable, such as f ''x 7. The rule can be justified in various ways. The derivative is the slope of the tangent to the given fu (such as 1) is zero; and, the product, of any number and zero, is zero.

4 Proof of the product rule

A rigorous proof of the product rule can be given using the properties of limitsIn mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets "close" to either some point, or infinity; or the behavior of a sequence's elements, as their index approaches infinity. Limits are used in calculu and the definition of the derivative as a limit of NewtonKneller's portrait of 1689. Sir Isaac Newton ( December 25, 1642 March 20, 1727 by the Julian calendar then in use; or January 4, 1643 March 31, 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemis's difference quotients:

Suppose

and suppose further that g and h are each differentiable at the fixed number x. Then

Since

we have

Since h is continuous at x, we have

and by the definition of the derivative, and the differentiability of h and g at x, we also have

Thus, we are justified in splitting each of the products inside the limit, and putting everything together, we have

and this completes the proof.

5 Generalizations

The product rule can be generalised to products of more than two factors. For example, for three factors we have

For a collection of functions , we can write this more succinctly as

It can also be generalized to higher derivatives of products of two factors: if y = uv and y⁽ⁿ⁾ denotes the n-th derivative of y, then

(see binomial coefficientIn mathematics, in particular in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number : and : (Here m denotes the factorial of m . The binomial coefficient of n and k is also written as C n). This result, often called the Leibniz rule, is formally quite similar to the binomial theoremIn mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. Its simplest version reads : whenever n is any non-negative integer and the numbers : are the binomial coefficients. This formula, and the triangular arra.

In multivariable calculus, the product rule is also valid for different notions of "product": scalar product and cross product of vectors, matrix product, inner products etc. All of these are summarized by the following general statement: let X, Y, Z be Banach spaces (which includes Euclidean space) and let B : X × Y → Z be a continuous bilinear operator. Then B is differentiable, and its derivative at the point (x,y) in X × Y is the linear map D_(x,y)B : X × Y → Z given by

6 See also

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1 Discovery by Leibniz