Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

An odd question (relationship between derivative and primitive)

  1. Nov 2, 2008 #1
    Is there an explicit formula for finding the antiderivative of a function? I was thinking that perhaps it would be the inverse function of the derivative, but I don't know what that would be off the top of my head.
  2. jcsd
  3. Nov 3, 2008 #2


    User Avatar
    Homework Helper

    There are (with some restrictions), but they are not useful

    one is the definite integral
    F=∫f dx
    another is differentiation can be effected by some opperator A
  4. Nov 3, 2008 #3


    User Avatar
    Staff Emeritus
    Science Advisor

    This is an example of the difference between "direct" and "inverse" methods.

    We have a specific definition of the "derivative" of a function and so, can, theoretically, find the derivative of any given (differentiable) function directly from that definition. The anti-derivative of the function f, on the other hand, is only defined as "the function that has f as its derivative". The first is a "direct problem" since we are given a direct definition and formula, the second is an "inverse problem" since the anti-derivative is only defined as an inverse of the derivative. "Inverse problems" are typically much harder than "direct problems".

    The same situation occurs in elementary algebra. If I define f(x)= x7- 3x5+ x4- 4x+ 5 and as "what is f(1)", that's easy: just set x= 1 and calculate: 1- 3+ 1- 4+ 5= 0, because I gave you the formula. If, instead, I ask you to solve the equation f(x)= 0, that is much harder- there is no general formula for solving such an equation. Having just calculated f(1)= 0 tells you that x= 1 is one solution, just as having learned that the derivative of x3 is 3x2 tells you that x3 is one anti-derivative of 3x2, but you still don't know if there are other solutions.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: An odd question (relationship between derivative and primitive)