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EXAMPLE 6Differentiating an Implicitly Defined Function

Find dydx if y=f(x) is defined implicitly by F(x,y)=eycosxx1=0.

Solution First we find the partial derivatives of F. Fx=Fx=eysinx1andFy=Fy=eycosx

Then we use (3). If eycosx0, dydx=FxFy=eysinx1eycosx=eysinx+1eycosx