A 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, there are clear counterexamples. Consider a differentiable function ƒ(x) whose derivative is ƒ '(x). This function can also be written as ƒ(x) · 1, since 1 is the identity element for multiplication. If the above-mentioned misconception were true, (u′)(v′) would equal zero. This is true because the derivative of a constant (such as 1) is zero and the product of ƒ '(x) · 0 is also zero.
Read more about this topic: Product Rule
Famous quotes containing the words common and/or error:
“Conscious virtue is the only solid foundation of all happiness; for riches, power, rank, or whatever, in the common acceptation of the word, is supposed to constitute happiness, will never quiet, much less cure, the inward pangs of guilt.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“There exists a black kingdom which the eyes of man avoid because its landscape fails signally to flatter them. This darkness, which he imagines he can dispense with in describing the light, is error with its unknown characteristics.... Error is certainty’s constant companion. Error is the corollary of evidence. And anything said about truth may equally well be said about error: the delusion will be no greater.”
—Louis Aragon (1897–1982)