Answer
$$\frac{\partial z}{\partial s} = 1582$$ $$\frac{\partial z}{\partial t} = 3164$$ $$\frac{\partial z}{\partial u} = - 700$$
Work Step by Step
1. According to the chain rule:
$$\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial s}$$ $$\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial t}$$ $$\frac{\partial z}{\partial u} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial u}$$
$$\frac{\partial z}{\partial x} = \frac{\partial (x^4 + x^2y)}{\partial x}= 4x^3 + 2xy$$ $$\frac{\partial z}{\partial y} = \frac{\partial (x^4 + x^2y)}{\partial y}= 0 + x^2 = x^2$$ $$\frac{\partial x}{\partial s} = \frac{\partial (s + 2t - u)}{\partial s} = 1$$ $$\frac{\partial x}{\partial t} = \frac{\partial (s + 2t - u)}{\partial t} = 2$$ $$\frac{\partial x}{\partial u} = \frac{\partial (s + 2t - u)}{\partial u} = -1$$ $$\frac{\partial y}{\partial s} = \frac{\partial (stu^2)}{\partial s} = tu^2$$ $$\frac{\partial y}{\partial t} = \frac{\partial (stu^2)}{\partial t} = su^2$$ $$\frac{\partial y}{\partial u} = \frac{\partial (stu^2)}{\partial u} = 2stu$$
Therefore:
$$\frac{\partial z}{\partial s} =(4x^3 + 2xy)(1) + (x^2)(tu^2)$$ $$\frac{\partial z}{\partial t} =(4x^3 + 2xy)(2) + (x^2)(su^2)$$ $$\frac{\partial z}{\partial u} =(4x^3 + 2xy)(-1) + (x^2)(2stu)$$
2. Calculate the value for each partial derivate.
$s = 4$
$t = 2$
$u = 1$
$x = s + 2t - u = (4) + 2(2) - (1) = 7$
$y = stu^2 = (4)(2)(1)^2 = 8$
$$\frac{\partial z}{\partial s} =((4(7)^3) + 2(7)(8)) \space (1) + ((7)^2)((2)(1)^2) = 1582$$ $$\frac{\partial z}{\partial t} =((4(7)^3) + 2(7)(8)) \space (2) + ((7)^2)((4)(1)^2) = 3164$$ $$\frac{\partial z}{\partial u} =((4(7)^3) + 2(7)(8)) \space (-1) + ((7)^2)(2(4)(2)(1)) = - 700$$
