The problem:
Find $\ Df(x,y,z)$ if $\ f(x,y,z) = (xz, y^2+z)$
I got confused, is it asking me for the domain in which the function is defined or about some sort of derivative?
$\endgroup$ 11 Answer
$\begingroup$Let $f(x,y,z)=(f_1(x,y,z),f_2(x,y,z))$ where $f_1,f_2:\Bbb R^3\to\Bbb R$ are the coordinate functions. Then $Df(x,y,z)$ is a type of derivative operator, and can be computed as follows: $$Df(x,y,z)=\begin{bmatrix}\frac{\partial f_1}{\partial x}&\frac{\partial f_1}{\partial y}&\frac{\partial f_1}{\partial z}\\\frac{\partial f_2}{\partial x}&\frac{\partial f_2}{\partial y}&\frac{\partial f_2}{\partial z}\end{bmatrix}$$ This derivative operator maps a vector function to a matrix consisting of all first order partial derivatives of the component functions of the input vector function. This matrix is known as the Jacobian of $f$. Note that, in your case, $f_1(x,y,z)=xz$ and $f_2(x,y,z)=y^2+z$.
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