The Jacobian of a transformation $u=u(x,y), \ v=v(x,y)$ is equal to
$$D=\text{det}\begin{pmatrix}\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\end{pmatrix}.$$
Show that if $f=u+iv$ is holomorphic then the Jacobian equals $|f'(z)^2|.$
Since a condition for holomorphicity is satisfaction of the Cauchy-Riemann equations we have that
$$D =\frac{\partial u}{\partial x} \frac{\partial v}{\partial y}-\frac{\partial u}{\partial y}\frac{\partial v}{\partial x}=\left(\frac{\partial u}{\partial x}\right)^2+\left(\frac{\partial v}{\partial x}\right)^2.$$
I want to compare this to $|f'(z)|^2,$ but how do I compute $f'(z)?$ According to $f=u+iv,$ $f$ does not seem to depend on $z$.
Is $z=u+iv?$
You're close!
By $f=u+iv$ they mean that $u$ and $v$ are real valued functions such that $f(z)=u(z)+iv(z)$. The $x$ and $y$ are for $z=x+iy$.