The problem states as follows:
Calculate the first order partial derivatives of the implicit function $z(x,y)$ that is defined by the equation $y^2 + e^{2xz}=\sin^{-1}(\frac{y}{z})$. Calculate their value at $(t,1)$, where $t$ is derived from the equation that defines $z(x,y)$ with $y=1, z=1$.
Ok I hope that made sense. (My teacher is a Russian immigrant and I translated this to English from Swedish).
The problem I have with this assignment is that, from my calculations, $t$ doesn't affect the value of the partials at the given point. So either I'm wrong or this is some sort of trick question. Can you spot some errors in my calculations?
EDIT: Since the partials don't depend on the value of t at the given point, this lead me to believe that this is some sort of trick question. (Or I'm wrong).
$$z_x=-\frac{2ze^{2xz}}{2xe^{2xz}+\frac{y}{z^2\sqrt{1-(\frac{y}{z})^2}}}$$ $$z_y=-\frac{2y-\frac{1}{z\sqrt{1-(\frac{y}{z})^2}}}{2xe^{2xz}+\frac{y}{z^2\sqrt{1-(\frac{y}{z})^2}}} $$ I simplified this to: $$z_y=-\frac{2yz\sqrt{1-(\frac{y}{z})^2}-1}{2xze^{2xz}\sqrt{1-(\frac{y}{z})^2}+\frac{y}{z}} $$ This doesn't lead to division by zero at $y=1, z=1$. This can't be said about $z_x$, which is undefined at that point. Eventhough any calculations of $t$ here are unnecessary, I got it to be: $$t= \frac{\ln(\frac{\pi}{2}-1)}{2} $$
My answer to this problem is that $z_x$ is undefined at $(t, 1, 1)$ and $z_y(t,1,1)=1$
Have I missed something? Thanks in advance