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:
“You know, what I very well know, that I bought you. And I know, what perhaps you think I don’t know, you are now selling yourselves to somebody else; and I know, what you do not know, that I am buying another borough. May God’s curse light upon you all: may your houses be as open and common to all Excise Officers as your wifes and daughters were to me, when I stood for your scoundrel corporation.”
—Anthony Henley (d. 1745)
“Never miss an opportunity to allow a child to do something she can and wants to on her own. Sometimes we’re in too much of a rush—and she might spill something, or do it wrong. But whenever possible she needs to learn, error by error, lesson by lesson, to do better. And the more she is able to learn by herself the more she gets the message that she’s a kid who can.”
—Polly Berrien Berends (20th century)