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:
“Commercial jazz, soap opera, pulp fiction, comic strips, the movies set the images, mannerisms, standards, and aims of the urban masses. In one way or another, everyone is equal before these cultural machines; like technology itself, the mass media are nearly universal in their incidence and appeal. They are a kind of common denominator, a kind of scheme for pre-scheduled, mass emotions.”
—C. Wright Mills (1916–62)
“Literature exists at the same time in the modes of error and truth; it both betrays and obeys its own mode of being.”
—Paul Deman (1919–1983)