How to find the derivative with respect to the transformed co-ordinates. I am stuck with something very simple , would be glad to get help . 
Suppose if i have a transformation matrix J , how do i find the derivative with respect to new co-ordinates , and derivative of function with respect to the transformed co-ordinates. 
For example i have a transformation



My interest is to find $dx_i'$ and $\frac{\partial f'}{\partial x_i'}$
$\mathbf{X}'=\mathbf{JX}\Rightarrow x'=x\cosh\theta+y\sinh\theta\Rightarrow dx'=\cosh\theta dx+\sinh\theta dy$

$\frac{\partial f'}{\partial x'}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial x'}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial y'}=\frac{f_{x}}{\sinh\theta}+\frac{f_{y}}{\cosh\theta}$

the same for $y$.
$\mathbf{X}'=\mathbf{JX}\Rightarrow x'=x\cosh\theta+y\sinh\theta\Rightarrow dx'=\cosh\theta dx+\sinh\theta dy$

$\frac{\partial f'}{\partial x'}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial x'}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial y'}=\frac{f_{x}}{\sinh\theta}+\frac{f_{y}}{\cosh\theta}$

the same for $y$.