The Quantity Z Can Be Expressed as a Function of X

Essay

The quantity z can be expressed as a function of x and y as follows: z = f(x, y).Now x and y are themselves functions of r and $\theta$ , as follows: $x = g ( r , \theta )$ and $y = h ( r , \theta )$ Suppose you know that g(1, $\pi$ /2)= -1, and h(1, $\pi$ /2)= 1.In addition, you are told that $$\begin{array} { l } \frac { \partial f } { \partial x } ( - 1,1 ) = 1 , \quad \frac { \partial f } { \partial y } ( - 1,1 ) = 6 , \quad \frac { \partial g } { \partial r } \left( 1 , \frac { \pi } { 2 } \right) = 7 \\\frac { \partial g } { \partial \theta } \left( 1 , \frac { \pi } { 2 } \right) = 7 , \quad \frac { \partial h } { \partial r } \left( 1 , \frac { \pi } { 2 } \right) = 6 , \quad \frac { \partial h } { \partial \theta } \left( 1 , \frac { \pi } { 2 } \right) = 4\end{array}$$ Find $\frac { \partial z } { \partial r } ( 1 , \pi / 2 )$