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:
“For Banquo’s issue have I filed my mind;
For them the gracious Duncan have I murdered;
Put rancors in the vessel of my peace
Only for them; and mine eternal jewel
Given to the common enemy of man,
To make them kings, the seeds of Banquo kings!”
—William Shakespeare (1564–1616)
“Custom calls me to’t.
What custom wills, in all things should we do’t,
The dust on antique time would lie unswept,
And mountainous error be too highly heaped
For truth to o’erpeer.”
—William Shakespeare (1564–1616)