Partial Differential equations/partial derivatives

Question

Suppose that $$f(x,y)=g(r,\theta)$$where $$x=\cosh \theta$$ $$y=r\sinh\theta$$

Find formulas for $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ entirely in terms of $r$, $\theta $, $\frac{\partial g}{\partial r} $ and $ \frac{\partial g}{\partial \theta}$.

Answer 1

Straight forward by differentiation :

Answer 2

Let $$f(x,y)=f(x(r,\theta),y(r,\theta))=g(r,\theta)$$ Now we have by the chain rule: $$\frac{\partial g}{\partial r}=\frac{\partial f}{\partial r}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial r}$$ and : $$\frac{\partial g}{\partial \theta}=\frac{\partial f}{\partial \theta}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial \theta}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial \theta}$$ Now differentiate x and y and solve the system.