Computing the jacobian and derivative of a function

Question

Let $f : \mathbb R^2 →\mathbb R^2$ defined by $f(x, y) := (xy, x + y)$.

Compute $J_f(v_0)$, the jacobian of $f$ at $v_0 ∈ \mathbb R^2$

Then compute $Df_{(1,1)} (h, h)$ [ Note: you may assume without proof that $f$ is differentiable at ($1, 1$) ]

My attempt:

Let $u=xy$ and $v=x+y$ then

$u_x = y$, $u_y = x$, $v_x = y$, $v_y = x$

So $J_f(v_0)$ = \begin{bmatrix}y&x\\ 1&1\end{bmatrix}

Is this correct?

But how do I compute $Df_{(1,1)} (h, h)$ ?

Answer 1

No it is not correct. Your partial derivatives of $v$ are wrong ! Correct is:

$$ v_x=v_y=1.$$

Hence

$$J_f(x,y)=\begin{bmatrix}y&x\\1&1\end{bmatrix}.$$