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

Reduced differentiation

  1. Apr 27, 2014 #1
  2. jcsd
  3. Apr 27, 2014 #2

    Mark44

    Staff: Mentor

    What are your thoughts on this?

    Given a function f, do you think it would be possible to express f'' as a single differentiation?
     
  4. Apr 27, 2014 #3
    Yeah!! Maybe, if is possible to find the nth antiderivative with an unique integral, so should be possible to find the nth derivative with an unique differentiation through of algebraic manipulation.
     
  5. Apr 27, 2014 #4

    lurflurf

    User Avatar
    Homework Helper

    There is Cauchy's differentiation formula
    $$\mathrm{f}^{(n)}(x)=
    \frac{n!}{2\pi \imath}\oint \frac{\mathrm{f}(z) \mathrm{d}z } {(z-x)^{n+1}}$$
    and some other related formula, but I do not recall any of the form
    $$\left( \dfrac{d}{dx}\right) ^n \mathrm{f}(x)=\dfrac{d}{dx} \mathrm{g}_n (x)\mathrm{f}(x)$$
    which given the rules of differentiation seems impossible for general f
    https://en.wikipedia.org/wiki/Cauchy's_integral_formula
     
  6. Apr 27, 2014 #5
    Yeah! I'm looking for something like this!
     
  7. Apr 28, 2014 #6
  8. Apr 28, 2014 #7

    Mark44

    Staff: Mentor

  9. Apr 28, 2014 #8
    It's more comprehensible express the derivative of a produt like way:
    $$(f\times g)^{(2)} = f^{(2)}g^{(0)} + 2f^{(1)}g^{(1)} + f^{(0)}g^{(2)}$$
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook