Take for example $f(x,y) = x^y$. I defined the total derivative to be the best linear approximation of $f$. Without working out the Jacobian I found that $$Df(x,y)(h_1,h_2) = h_1yx^{y-1} + h_2x^y\log(x)$$
However the Jacobian gives me a $1 \times 2$ matris: $$\begin{bmatrix} yx^{y-1} & x^y \log (x) \end{bmatrix} $$
I don't really understand how computing the total derivative explicitly gives me a real number, and computing the Jacobian gives me a matrix, how are they related because I do know they somehow are.
Best Answer
The Jacobian is just a linear function. Apply it to the point $\begin{bmatrix} h_1 \\ h_2\end{bmatrix}$ and you get back the total derivative.