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:
“Mankind’s common instinct for reality ... has always held the world to be essentially a theatre for heroism. In heroism, we feel, life’s supreme mystery is hidden. We tolerate no one who has no capacity whatever for it in any direction. On the other hand, no matter what a man’s frailties otherwise may be, if he be willing to risk death, and still more if he suffer it heroically, in the service he has chosen, the fact consecrates him forever.”
—William James (1842–1910)
“Error is to truth as sleep is to waking. I have observed that one turns, as if refreshed, from error back to truth.”
—Johann Wolfgang Von Goethe (1749–1832)