I'm working on a proof for the local truncation error of the implicit Euler method. I've been given a little hint to get started, but I'm stuck on a line that involves the chain rule. Can some please explain to me how this works out? Given:
$\frac{dy}{dt} = f(y),\;\;y(0) = y_{0},\;\;$ and $u_{n+1} = u_{n} + hf(u_{n+1})$
If I have $$f(y_{n+1}) = f(y_n) + f_{,y}(y_{n+1}-y_n) + higher\,order\,terms $$
How does that become $$f(y_{n+1}) = f(y_n) + f_{,y}\,hy' + O(h^2) $$ Specifically, how do you get the $\,f_{,y}\,hy'\,$?