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 fatuously hoped that we might pluck from the human tragedy itself a consciousness of a common destiny which should bring its own healing, that we might extract from life’s very misfortunes a power of cooperation which should be effective against them.”
—Jane Addams (1860–1935)
“We call contrary to nature what happens contrary to custom; nothing is anything but according to nature, whatever it may be, Let this universal and natural reason drive out of us the error and astonishment that novelty brings us.”
—Michel de Montaigne (1533–1592)