12. Suppose that f(x, y, z. w) = 0 and g(x, y, z, w) = 0 determine z and w as differentiable functions of the independent variables x and y, and suppose that \frac{\partial f}{\partial z} \frac{\partial g}{\partial w}-\frac{\partial f}{\partial w} \frac{\partial g}{\partial z} \neq 0 Show that \left(\frac{\partial z}{d x}\right)_{y}=-\frac{\frac{\partial f}{\partial x} \frac{\partial g}{\partial v}-\frac{\partial f}{\partial w} \frac{\partial g}{\partial x}}{\frac{\partial f}{\partial z} \frac{\partial g}{\partial h}-\frac{\partial f}{\partial h v} \frac{\partial g}{d z}} and \left(\frac{\partial w}{\partial y}\right)_{\bar{x}}=-\frac{\frac{\partial f}{d z} \frac{\partial g}{\partial y}-\frac{\partial f}{\partial y} \frac{\partial g}{d z}}{\frac{\partial f}{\partial z} \frac{\partial g}{\partial w}-\frac{\partial f}{\partial w} \frac{\partial g}{\partial z_{z}}} .

Solved: 12. Suppose that f(x, y, z. w) = 0 and g(x, y, z, w) = 0 determine z

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12 suppose that fx y z w 0 and gx y z w 0 determine z and w as differe

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12. Suppose that f(x, y, z. w) = 0 and g(x, y, z, w) = 0 determine z and w as differentiable functions of the independent variables x and y, and suppose that \frac{\partial f}{\partial z} \frac{\partial g}{\partial w}-\frac{\partial f}{\partial w} \frac{\partial g}{\partial z} \neq 0 Show that \left(\frac{\partial z}{