Suppose that $f$ is a function of three independent variables $x, y$, and $z .$ Show that $D_{mathrm{i}} f=f_{x}, D_{mathrm{j}} f=f_{y}$, and $D_{mathrm{k}} f=f_{z}...

Suppose that the equation $g(x, y, z)=0$ determines $z$ as a differentiable function of the independent variables $x$ and $y$ and that $g_{z} \neq 0 .$ Show that
$$\left(\frac{\partial z}{\partial y}\right)_{x}=-\frac{\partial g / \partial y}{\partial g / \partial z}.$$

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So in this video are asked to prove that D. C by the x Times dx spike divided times Do I buy disease? Negative. So I asked the profits right here. So what we can do issues the implicit formulas that we know so we can write easy by TXs negative f sub x way of subsidy times DX Mikey wise Negative f sub live light of setbacks times now do you see by The divide by Deasy is negative of subsidy divided by a subway when we noticed them we can cancel all of these and these two negatives cancel we're left with is negative.

This video covers implicit differentiation and partial derivatives. And this problem, we are given that F is a function of X. Y and Z. Where X, Y and Z are all functions of the other two variables. You are also given that the partial derivative of F with respect to X, Y and Z are all not zero. You asked to prove that the partial derivative of Z with respect to X times the partial derivative effects with respect to Y times the partial derivative Of why with respect to Z is equal to -1. We have equations for all of these partial derivatives. In terms of F sub X, F, sub Y and F sub C. These are at the partial derivative of Z with respect to X is equal to the partial derivative of F with respect to X. Over the partial derivative of F with respect to Z. The negative of that. The partial derivative facts with respect to Y is equal to negative of the partial derivative of F with respect to Y over F. With respect to X. And the partial derivative of Why with respect to see is the negative of the partial derivative of have with respect to Z over F. With respect to Y. You can now combine all of these to find that the partial derivative of Z with respect to X times the partial derivative that's with respect to Y. China's. The partial derivative of Y with respect to Z is equal to what is written here and after we have written that we can cancel out. Yeah. Until we or just left with, Can I get 1? And that is our

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