# 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:
$$(f\times g)^{(2)} = f^{(2)}g^{(0)} + 2f^{(1)}g^{(1)} + f^{(0)}g^{(2)}$$