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:
“Down these mean streets a man must go who is not himself mean, who is neither tarnished nor afraid.... He is the hero, he is everything. He must be a complete man and a common man and yet an unusual man. He must be, to use a rather weathered phrase, a man of honor, by instinct, by inevitability, without thought of it, and certainly without saying it. He must be the best man in his world and a good enough man for any world.”
—Raymond Chandler (1888–1959)
“It is the very error of the moon,
She comes more near the earth than she was wont,
And makes men mad.”
—William Shakespeare (1564–1616)