How do I express the second derivative of this function

Question

We have $$F(X)=f(x)\psi(x)$$ $$X=\chi(x)$$

I can show that $$\frac{dF}{dX}=\frac{1}{\chi^\prime}(\psi f^\prime+f \psi^\prime)$$

But don't know how to express

$$\frac{d^2F}{dX^2}=\text{?}$$

Answer 1

You differentiate a second time what you have already calculated

$$\frac{d^2F}{dX^2}=\frac{d}{dX}\frac{dF}{dX}=\left (\frac{d}{dx}\left(\frac{dF}{dX}\right )\right )\frac 1{\chi^\prime}$$

Where $$\frac{dF}{dX}=\frac{\psi y^\prime+y \psi^\prime}{\chi^\prime}$$ Therefore $$\frac{d^2F}{dX^2}=\frac 1{\chi^\prime}\left (\frac{d}{dx}\left(\frac{\psi y^\prime+y \psi^\prime}{\chi^\prime}\right )\right )$$

Note that $$\chi'=\frac {dX}{dx}$$