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:
“Nothing is more common than mutual dislike, where mutual approbation is particularly expected.”
—Samuel Johnson (1709–1784)
“In religion,
What damned error but some sober brow
Will bless it, and approve it with a text,
Hiding the grossness with fair ornament?”
—William Shakespeare (1564–1616)