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Find the partial derivative of the dependent variable or function with respect to each of the independent variables.$$u=\ln \frac{y^{2}}{x-y}+e^{-x}(\sin y-\cos 2 y)$$

Calculus 3

Chapter 29

Partial Derivatives and Double Integrals

Section 3

Partial Derivatives

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All right. We want to find the first partial derivatives of the function F of x Y is equal to Ellen, Y squared over x minus Y plus even the negative X times sine Y minus co signed to. Why? This question is challenging our understanding of differentiation as applied to multi variant functions. The first partial results were searching for our fx or DF dX and dY we use standard differentiation techniques. We learned in single variable calculus to find both, but for DF dx, which requires a constant. And for the FBI we treat X as a constant so that we can proceed to solve fx is three of Ellen Y squared over x minus Y with respect to x x minus Y over wide square times negative Y squared over x minus Y squared minus. Either negative X and y minus two Coast Costa Y. Or simplifying negative one over X y minus even em fx sign Y minus Costa Y. Then F Y is by a similar process of chain rule on an algorithm one over X minus Y plus two over Y, plus either negative X. To sign two Y plus coast Y as indicated the bottom.

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