Analysis: Finding the derivative of a two variable function.

Question

Question: find the derivative of $f(x,y)=(cos(x)cosh(y),sin(x)sinh(y))$ at the point (p,q).

I am fairly stuck here as each variable is in each component so I cant simply take the partial of each component and say that that is the derivative. So I am not sure how to proceed, any hints or advice would be appreciated.

Answer 1

Let $f_1(x,y)=\cos (x) \cosh (y)$ and $f_1(x,y)=\sin (x) \sinh (y)$ . Then

$f'(p,q)$ is the Jacobi-matrix of $(f_1,f_2)$.