Let $f(x,y,z)=(x^2\cos (yz), \sin (x^2y)-x, e^{y \sin z})$. Compute the derivative matrix $Df$.

Let $f(x,y,z)=(x^2\cos (yz), \sin (x^2y)-x, e^{y \sin z})$. Compute the derivative matrix $Df$.

Answer

We have 

\[\begin{pmatrix} 2x \cos (yz)& -x^2z \sin (yz) & -x^2 y \sin (yz) \\ 2x y \cos (x^2y) -1& x^2\cos (x^2y) & 0 \\ 0 & \sin z e^{y \sin z} & y\cos z e^{y \sin z} \end{pmatrix}. \]
Note that 

- the firts row is partial derivatives of $x^2 \cos (yz)$ with respect to $x,y,$ and $z$, respectively. 
- the second row is partial derivatives of $\sin (x^2y)-x$ with respect to $x,y,$ and $z$, respectively. 
- the third row is partial derivatives of $e^{y\sin z}$ with respect to $x,y,$ and $z$, respectively. 

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