Partial derivative multivariate

Question

this operation is making me crazy. Can someone help me please?

$ g(h) := f(t + h, u + hf(t, u(t))$, with $u'(t) = f(t, u(t))$

So what is $g'(h)? $ Thank you

Answer 1

Denote $f_t$ be partial derivative of $f(t,u)$ w.r.t the first variable, and $f_u$ be the partial derivative w.r.t the second variable,

Note $g(h)$ is only a function of $h$, so $t$ can be viewed as a constant, so is $u(t),f(t,u(t))$. Hence by chain rule,

$$ g'(h)=f_t(t + h, u + hf(t, u(t))+f_u(t + h, u + hf(t, u(t))f(t,u)$$