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:
“All the philosophy, therefore, in the world, and all the religion, which is nothing but a species of philosophy, will never be able to carry us beyond the usual course of experience, or give us measures of conduct and behaviour different from those which are furnished by reflections on common life. No new fact can ever be inferred from the religious hypothesis; no event foreseen or foretold; no reward or punishment expected or dreaded, beyond what is already known by practice and observation.”
—David Hume (1711–1776)
“In Pride, in reas’ning Pride, our error lies;
All quit their sphere, and rush into the skies.
Pride still is aiming at the blest abodes,
Men would be Angels, Angels would be Gods.”
—Alexander Pope (1688–1744)