This question is taken from Vector Calculus by Marsden and Tromba. It is question 39 of Page 231. It reads as follows:
Consider the system of equations \begin{align*} 2xu^3v - yv&=1 \\ y^3v+x^5u^2&=2 \end{align*} Show that near the point $(x,y,u,v)=(1,1,1,1)$, this system defines $u$ and $v$ implicitly as functions of $x$ and $y$. For such local functions $u$ and $v$, define the local function $f$ by $f (x, y) = (u(x, y), v(x, y))$. Find $D f (1, 1)$.
I have managed to prove the first part by using the Implicit Function Theorem. I am struggling with the second part, about finding the derivative matrix. I know that $$Df(x,y) = \begin{bmatrix}u_x & u_y \\ v_x & v_y \end{bmatrix}$$ and so the question wants me to find the partials.
I do so by differentiating the given equations wrt $x$ and $y$, obtaining $$\begin{align*}2u^3 v + 6xu^2vu_x + 2xu^3 v_x - yv_x&=0 \\ 6xu^2vu_y + 2xu^3v_y-v-yv_y&=0 \\ y^3 v_x + 5x^4u^2 + 2x^5u u_x &=0 \\ 3y^2x+y^3v_y + 2x^5uu_y &= 0 \end{align*}$$
Setting $(x,y,u,v) = (1,1,1,1)$, I get $$6u_x+v_x=-2 \\ 6u_y + v_y = 1 \\ 2u_x + v_x = -5 \\ v_y + 2u_y = -3 $$ which when solving I get $u_x = 3/4, v_x = -13/2, u_y = 1, v_y =-5$. However, the given answer is $$\begin{bmatrix} -7/4 & 1 \\ 17/2 & -5 \end{bmatrix}$$ which means that my partials wrt $x$ are wrong.
Please, could someone help me find where I went wrong? Thank you very much.