I've recently been studying some basic calculus, and happened to find somewhat of a paradox.
Let's consider the three following functions :
$f(x) = 2x+5$
$g(x) = 3x-1$
$h(x) = f(g(x)) = 6x+3$
The differentials for the first two functions are :
$f'(x) = 2$
$g'(x) = 3$
However once we consider the differential for $h(x)$, things start to break down ...
If we use the chain rule form, we get :
$h'(x) = g'(x)\times f'(g(x)) = 3 \times 2 \times (3x-1) = 18x-6$
However, if we use the simpler linear form :
$h'(x) = 6$
Can somebody explain in simple, intuitive terms why using the chain rule form in this context does not work ?
By extension, what are the limitations and conditions in order to use the chain rule in calculus ?