Expert: Paul Klarreich - 10/7/2005

Question
Could you please explain what an antiderivative of a derivative is?  Thanks.  

Answer
Hi, Kim,
The antiderivative is (almost) the opposite of differentiation.  Given a function f(x), if you differentiate it, you get a new function, usually denoted f'(x).  

The antiderivative of a function essentially solves a puzzle:  Given a function, denoted f'(x), that is the derivative of some other function, what was that other function?

So if we take the derivative of f(x), and then the antiderivative of the result, we get back (almost) the original function.

Why almost?  Suppose f(x) = x^2.  Then f'(x) = 2x.
The antiderivative of f'(x) = 2x means to find a function whose derivative is 2x.  

x^2 is such a function, and it is the original.  But so is  x^2 + 6,
and so is x^2 - 5, and in fact any function of the form x^2 + C will work.