Suppose I have $ax +b =c$. I can solve $\partial x/\partial c$ by IFT assuming that $a,b$ are constants and $x,c$ are variables: let $F= ax +b -c$. Then, $$\partial x/\partial c = -F_c/F_x =1/a$$
Now it's also true that $e^{ax +b}= e^c $, but when I apply IFT to $F'=e^{ax +b}- e^c$, I get the wrong derivative:
$$\partial x/\partial c = -F'_c/F'_x =\frac{e^c}{a e^{ax+b}} \neq 1/a$$
Can you tell me where am I making a mistake in the calculations?
A montone transformation of both sides of the original equality should not change the result.
So thanks to lulu's comment I solved my doubt.
Indeed $e^c = e^{ax + b}$ by my initial equality. So $-F'_c/F'_x = 1/a.$
A bit more generally, knowing that $$f(ax + b) = f(c),$$ we can compute $\partial x/ \partial c$ by applying IFT to $F' =f(ax + b) - f(c)$:
$$ \partial x/ \partial a = - \frac{-f'(c)}{f'(ax + b) a}= 1/a. $$
$f'(c) = f'(ax + b) $ follows by differentiating both sides of the first displayed equality.