Question

Let z = f(x, y), x = x(u, v), y = y(u, v) and x(1, 2) = 4, y(1, 2) = 3, calculate the partial derivative in terms of some of the numbers a, b, c, d, m, n, p, q. fx(1, 2) = a fy(1, 2) = c xu(1, 2) = m yu(1, 2) = p fx(4, 3) = b fy(4, 3) = d xv(1, 2) = n yv(1, 2) = qzu(3,4)=_________

Answer 1

To summarize, we're given

$\begin{cases}z=f(x,y)\\x=x(u,v)\\y=y(u,v)\\x(1,2)=4\\y(1,2)=3\end{cases}$

and we're asked to find the value of $z_u(4,3)$ (I think there's a typo in your question) given that

$\begin{cases}f_x(1,2)=a\\f_y(1,2)=c\\x_u(1,2)=m\\y_u(1,2)=p\\f_x(4,3)=b\\f_y(4,3)=d\\x_v(1,2)=n\\y_v(1,2)=q\end{cases}$

By the chain rule, we have

$\dfrac{\partial z}{\partial u}=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial u}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial u}$

or in subscript notation,

$z_u=z_xx_u+z_yy_u$

Then

$z_u(4,3)=z_x(4,3)x_u(1,2)+z_y(4,3)y_u(1,2)=\boxed{bm+dp}$