Problem:
Find the jacobian of the vector valued function $\begin{pmatrix} f(x,y,z)\\ g(x,y,z) \\ h(x,y,z)\end{pmatrix}$ where $f(x,y,z) = \tan \left(x^3\right)$, $g(x,y,z) = \frac{1}{\sqrt{y}}$, and $h(x,y,z) = \sin ^{-1}\left(\frac{y}{x^3}\right)$
Answer:
$\left(
\begin{array}{ccc}
 3 x^2 \sec ^2\left(x^3\right) & 0 & 0 \\
 0 & -\frac{1}{2 y^{3/2}} & 0 \\
 -\frac{3 y}{x^4 \sqrt{1-\frac{y^2}{x^6}}} & \frac{1}{x^3 \sqrt{1-\frac{y^2}{x^6}}} & 0 \\
\end{array}
\right)$
{'source': 'AMPS', 'file': 'amps/mathematica/calculus/jacobian/36768.txt'}