So we have $f(x,y)$ which is a function in $R^2$, we have $x = g(t_1, t_2)$ and $y = h(t_1, t_2$). My task is to find the partial derivatives to: $$u(t_1, t_2) = f(g(t_1,t_2), h(t_1,t_2))$$
I am supposed to find $u^{'}_1$ and $u^{'}_2$ in the point $(1,1)$ using the following table:
So my approach to the problem is to simply calculate the partial derivatives for $$\frac{\partial u}{\partial t_1} = \frac{\partial f}{\partial x} = \frac{\partial f}{\partial g}\frac{\partial g}{\partial t_1} + \frac{\partial f}{\partial h}\frac{\partial h}{\partial t_1}$$
If i evaluate this expression at $(1,1)$ using the table above I get the result $5 * 2 + (-2) * 2 = 6$. The correct answer should be $10$. I get another error if I calculate $\frac{\partial u}{\partial t_2}$.
Any help would be much appreciated.
You are evaluating the derivatives of $f$ at the wrong point. You need to evaluate them at $(1,2)$ not $(1,1),$ because $g(1,1)=1, h(1,1)=2$.