# Math Genius: Relating the total derivative and the Jacobian matrix

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^ylog(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.

The Jacobian is just a linear function. Apply it to the point \$begin{bmatrix} h_1 \ h_2end{bmatrix}\$ and you get back the total derivative.

The total derivative is the best linear approximation of \$f\$, thus an affine plane in \$x,y\$ space.

\$\$
f(x+h_1,y+h_2) = f(x, y) + mbox{grad } f cdot (h_1, h_2)^T + O(h^2)
\$\$

Compare this with the \$f’ : mathbb{R}^2 to mathbb{R}\$ in
\$\$
lim_{(h_1, h_2) to 0}
frac{lVert f(x+h_1, y+h_2) – f(x,y) – f'(x,y) (h_1, h_2)^TrVert}{lVert (h_1, h_2)^T rVert} = 0
\$\$
You computed
\$\$
Delta f approx mbox{grad } f cdot (h_1, h_2)^T
\$\$
while the total derivative is
\$\$