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:
“... a nation to be strong, must be united; to be united, must be equal in condition; to be equal in condition, must be similar in habits and feeling; to be similar in habits and feeling, must be raised in national institutions as the children of a common family, and citizens of a common country.”
—Frances Wright (1795–1852)
“If the individual, or heretic, gets hold of some essential truth, or sees some error in the system being practised, he commits so many marginal errors himself that he is worn out before he can establish his point.”
—Ezra Pound (1885–1972)