Reduced differentiation

1. Apr 27, 2014

Jhenrique

2. Apr 27, 2014

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?

3. Apr 27, 2014

Jhenrique

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.

4. Apr 27, 2014

lurflurf

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

5. Apr 27, 2014

Jhenrique

Yeah! I'm looking for something like this!

6. Apr 28, 2014

bigfooted

7. Apr 28, 2014

Staff: Mentor

8. Apr 28, 2014

Jhenrique

It's more comprehensible express the derivative of a produt like way:
$$(f\times g)^{(2)} = f^{(2)}g^{(0)} + 2f^{(1)}g^{(1)} + f^{(0)}g^{(2)}$$