How do I perform taylor expansion of the following?

Question

Taylor expansion about $(x,y)$ of $f(x + a,\; y + k\; f(x + b,\; y + c))$

I do not understand what happens to the second $f$ inside. The inspiration for this question is Runge-Kutta methods.

Answer 1

Given $f(s,t)$, define $g(x,y) = f(x+a, y+kf(x+b,y+c))$.

Then partial derivatives of $g$ are, by chain rule, $$\partial_x g (x,y)= (\partial_s f + k \partial_t f \partial_s f)(x+a, y+kf(x+b,y+c))$$ $$\partial_y g (x,y)= (k \partial_t f \partial_t f)(x+a, y+kf(x+b,y+c))$$

We obtain a first order Taylor's expansion. Higher order expansion can be done in a similar way.