# 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 $\frac{\partial f'}{\partial x_i'}$

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$\mathbf{X}'=\mathbf{JX}\Rightarrow x'=x\cosh\theta+y\sinh\theta\Rightarrow dx'=\cosh\theta dx+\sinh\theta dy$
$\frac{\partial f'}{\partial x'}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial x'}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial y'}=\frac{f_{x}}{\sinh\theta}+\frac{f_{y}}{\cosh\theta}$
the same for $y$.
Can you tell me what $dx'$ means here and how you find it \? –  Theorem Oct 25 '12 at 22:38
$dx_1=dx$ and $dx_2=dy$ and $(\sinh\theta)'=\cosh\theta$ , $dx'$ is the change of x in the new coordinates (prime in all those equations is denoting "new coordinates" not derivative) –  TMS Oct 25 '12 at 22:43
Why have you got $dx'=sinh\theta dx + ycosh\theta dy$ ?? something that don't understand :( –  Theorem Oct 26 '12 at 5:42