The implicit function theorem gives conditions under the relationship G(x,y)=0 defines y implicitly as a function of x. Use the implicit function theorem to show that the relationship x+y+e^{xy}=0 defines y implicitly as a function of xnear the point (0,-1).

The implicit function theorem gives conditions under the relationship G(x,y)=0 defines y implicitly as a function of x. Use the implicit function theorem to show that the relationship x+y+e^{xy}=0 defines y implicitly as a function of xnear the point (0,-1).
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Let F(x,y,) be a continuous function with continuous partial derivative defined on an open set S containing the point $P=\left({x}_{0},{y}_{0}\right)$. If $\mathrm{\partial }F/\mathrm{\partial }y\ne 0$ at P then there exists a region R about ${x}_{0}$ such that for any x in R there is a unique y such that F(x,y)=0
Here $F\left(x,y\right)=x+y+{e}^{xy}$ is continuous andat the point P(0.-1) the partial derivative $\mathrm{\partial }F/\mathrm{\partial }y=1+x{e}^{xy}$ at (0,-1) is nonzero.
Hence from the Implicit function theorem, we can say that ,there exists a function in some neighbouhoodof (0,-1) such that y=f(x).
One possible entry for neighbourhood is{(x,y):|(x,y)-(0,-1)|<1} in which $\mathrm{\partial }F/\mathrm{\partial }y\ne 0$