A general form derivative problem

[EngWiPy]

Hello,
Suppose that $$f(x)=g\left(h(x)\right)$$, then can we write $$f^{(n)}(x)=g^{(n)}\left(h(x)\right)\times k\left(h(x)\right)$$??
Note: $$f^{(n)}(x)=\frac{d^n\,f(x)}{dx^n}$$.
Regards

 

[mathman]

Hello,
Suppose that $$f(x)=g\left(h(x)\right)$$, then can we write $$f^{(n)}(x)=g^{(n)}\left(h(x)\right)\times k\left(h(x)\right)$$??
Note: $$f^{(n)}(x)=\frac{d^n\,f(x)}{dx^n}$$.
Regards

No.
f'(x)=g'(h(x))h'(x)
f''(x)=g''(h(x))[h'(x)]²+g'(h(x))h''(x)

It doesn't look anything like what you are proposing.

 

[Vid]

http://en.wikipedia.org/wiki/Faà_di_Bruno's_formula

It's a fairly involved formula.

 

[EngWiPy]

Thanks for replying. Let me to be more specific now, I think it may simplify the general form as given by Vid. I want to find the $$n^{th}$$ derivative with repect to $$s$$of the following function:

$$\frac{\Psi(a,b;\varphi^-(s))-\Psi(a,b;\varphi^+(s))}{\sqrt{(\zeta-s)^2-4\beta^2}}$$​

where

$$\varphi^{\pm}=(\zeta-s)\pm \sqrt{(\zeta-s)^2-4\beta^2}$$​

and

$$\frac{\partial^n}{\partial z^n}\Psi(a,b;z)=(-1)^n\,(a)_n\Psi(a+n,b+n;z)$$​

where

$$(a)_n=a(a+1)\ldots (a+n-1)$$​