Development of this rule is credited to Leibniz; who demonstrated that (x + dx)(y + dy) - xy = x(dy) + y(dx); as (dx)(dy) is "negligible". If two differentiable functions u and v of the variable x are given, then the derivative of their product uv equals the derivative of u, multipled by v; and added to u, multiplied by the derivative of v:
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 it derivative equals c multiplied by the derivative of f(x). (This follows from the product rule since the derivative of any constant is zero.)
The product rule can be used to derive the rule for integration by parts and the quotient rule.
Table of contents |
2 Informal Proof 3 Informal justification of the product rule 4 Generalizations |
It is a common misconception, when studying calculus, to suppose that the derivative of (uv) equals (u')(v'); 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 be also written f(x) · 1, since 1 is the identity element for multiplication. Now, suppose the above mentioned misconception were true; if so, (u')(v') would equal zero; since, the derivative of a constant (such as 1) is zero; and, the product, of any number and zero, is zero. Such a misconception could result in a belief that the derivative, of every function, is 0; such a belief is not correct.
If
The product rule can be justified as follows. Let:
Let the independent variable x increase by a small amount Δx, resulting in an increase of u by Δu, of v by Δv, and of y by Δy. Now:
In multivariable calculus, the product rule is also valid for different notions of "product": scalar product and cross product of vectorss, 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