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:
“We early arrive at the great discovery that there is one mind common to all individual men: that what is individual is less than what is universal ... that error, vice and disease have their seat in the superficial or individual nature.”
—Ralph Waldo Emerson (1803–1882)
“Error is a supposition that pleasure and pain, that intelligence, substance, life, are existent in matter. Error is neither Mind nor one of Mind’s faculties. Error is the contradiction of Truth. Error is a belief without understanding. Error is unreal because untrue. It is that which seemeth to be and is not. If error were true, its truth would be error, and we should have a self-evident absurdity—namely, erroneous truth. Thus we should continue to lose the standard of Truth.”
—Mary Baker Eddy (1821–1910)