I am studying the relationship between two similar functions:
$$ F(g(x)) = k_1 \int_{0}^x g(x) f(x) dx$$ $$ F(h(x)) = k_2 \int_{0}^x h(x) f(x) dx$$
I am interested in properties of the graph constructed using the first function as the x coordinate and the second function as the y coordinate. So I am interested in calculating derivatives of F(h(x)) with respect to F(g(x)).
I think that I have the first derivative:
$$ \frac{dF(h(x))}{dF(g(x))} = \frac{k_2h(x)}{k_1g(x)}$$
I derived this with the chain rule and the fundamental thm of calculus.
It shouldn't be that difficult, but I can't seem to calculate the second derivative. I found a source that suggests that the answer should be
$$ \frac{d^2F(h(x))}{dF^2(g(x))} = \frac{k_2h(x)}{k_1^2(g(x))^2} \frac{\alpha_h-\alpha_g}{xf(x)}$$
where
$$ \alpha_g = \frac{dg}{dx}\frac{x}{g}; \alpha_h = \frac{dh}{dx}\frac{x}{h} $$
Any/all assistance much appreciated. Thanks, Eric