Assuming [tex]z=z(x,y)[/tex], differentiating both sides wrt [tex]x[/tex] gives
[tex]2x+14z\dfrac{\partial z}{\partial x}=0\implies\dfrac{\partial z}{\partial x}=-\dfrac x{17z}[/tex]
and wrt [tex]y[/tex] gives
[tex]8y+14z\dfrac{\partial z}{\partial y}=0\implies\dfrac{\partial z}{\partial y}=-\dfrac4{7z}[/tex]
The required differential functions ∂z/∂x and ∂z/∂y are -2x/14z and -8y/14z
Given the equation:
x² + 4y² + 7z² = 1
Rewriting the equation
x² + 4y² + 7z² - 1 = 0
f(x, y, z) = 0
We are to find the differential equations ∂z/∂x and ∂z/∂y
∂z/∂x means we are to differentiate with respect to x keeping y constant
Differentiating using the implicit differentiation
Recall
x² + 4y² + 7z² - 1 = 0
On differentating with respect to x:
2x + 0 + 14x ∂z/∂x = 0
14z ∂z/∂x = - 2x
14z ∂z/∂x = -2x
∂z/∂x = -2x/14z
Similarly for ∂z/∂y
Differentiating implicitly with respect to y;
0 + 8y + 14z∂z/∂y = 0
14z∂z/∂y = -8y
∂z/∂y = -8y/14z
Hence the required differential functions ∂z/∂x and ∂z/∂y are -2x/14z and -8y/14z
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