Question 7.1.4: Suppose y is defined implicitly as a function of x by the eq......

Essential Mathematics for Economic Analysis [1185377]

Suppose y is defined implicitly as a function of x by the equation

$g(x y^{2})=x y+1$ (∗)

where g is a given differentiable function of one variable. Find an expression for $y^{\prime}.$

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We differentiate each side of the equation w.r.t. x, considering y as a function of x. The derivative of g(xy²) w.r.t. x is $g^{\prime}(xy^{2})(y^{2} + x^{2}yy^{\prime}).$ So differentiating (∗) yields $g^{\prime}(xy^{2})(y^{2} + x^{2}yy^{\prime}) = y + xy^{\prime}.$ Solving for $y^{\prime}$ gives us

y^{\prime}=\frac{y\left[y g^{\prime}(x y^{2})-1\right]}{x\left[1-2y g^{\prime}(x y^{2})\right]}

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