Using Taylor's expansion, show that for every $t\in[t_0,T]$ it is true that: $$f(t,y(t))\ \approx\ f(t_k,y_k) + (t-t_k)f_t(t_k,y_k) + (t-t_k)f(t_k,y_k)f_y(t_k,y_k)\,.$$
For this problem we have that \begin{equation}\label{eqn:ode:1} \left\{\begin{array}{rcl} y^{\prime} & = & f(t,y)\,,\; \text{for } t_0\leq t\leq T\\[0.5ex] y(t_0) & = & y_0 \end{array}\right. \end{equation}
The only thing that occurs to me is to use Taylor for several variables, that is $$ f(t,y(t))=f(t_k,y(t_k)) +(t-t_k)f_t(t_k,y(t_k))+(t-t_k)f_y(t_k,y(t_k)) + \cdots$$
however, I have no idea how to proceed, someone could help me?
Let \begin{align} g(t):= f(t, y(t)) \end{align} then Taylor expanding $g(t)$ about $t=t_k$ yields \begin{align} g(t) = g(t_k) + g'(t_k)(t-t_k)+\mathcal{O}(|t-t_k|^2) \end{align} when $|t-t_k|$ is sufficiently small. In particular, we see that \begin{align} g'(t) =&\ \partial_t f(t, y(t))+\partial_yf(t, y(t))y'(t)\\ =&\ \partial_t f(t, y(t))+\partial_yf(t, y(t))f(t, y(t)) \end{align} Hence we have the desired result.