Find the derivative of each implicit function, where dY/dX = -F_X/F_Y, provided that F_Y does not...

Question:

Find the derivative of each implicit function, where {eq}\frac{dY}{dX} = -\frac{F_{X}}{F_{Y}} {/eq}, provided that {eq}F_{Y} \neq 0 {/eq}.

a) {eq}F(x,y) = x^{2} + y^{2} + (xy)^{\frac{1}{3}} = 0 {/eq}

b) {eq}F(x,y) = x^{2} y + y^{2}x + xy = 0 {/eq}

Implicit Differentiation:

Implicit differentiation uses the chain rule to differentiate a function that is implicitly defined. Most of the problems are written as explicit functions of x. For example, {eq}y = 2x^2 + 4 {/eq}. For explicitly written functions, it is easy to find the derivatives of the function y with respect to the variable x. However, some functions in economics are written implicitly as functions of x and y. For example, {eq}U(x,y) = x^{1/3}y^{2/3} {/eq}. For this function, the derivative of the function y with respect can only be differentiated using the chain rule.

a) {eq}F(x,y) = x^{2} + y^{2} + (xy)^{\frac{1}{3}} = 0 {/eq}

The partial derivative of F(x,y) with respect to x is:

{eq}F_x = 2x + \frac{1}{3}(y)^{\frac{-2}{3}} {/eq}

The partial derivative of F(x,y) with respect to y is:

{eq}F_y = 2y + \frac{1}{3}(x)^{\frac{-2}{3}} {/eq}

Therefore:

{eq}\frac{dY}{dX} = -\frac{F_{x}}{F_{y}} = \displaystyle -\frac{2x + \frac{1}{3}(y)^{\frac{-2}{3}}}{2y + \frac{1}{3}(x)^{\frac{-2}{3}} } {/eq}

b) {eq}F(x,y) = x^{2} y + y^{2}x + xy = 0 {/eq}

The partial derivative of F(x,y) with respect to x is:

{eq}F_x = 2xy + y^2 + y {/eq}

The partial derivative of F(x,y) with respect to y is:

{eq}F_y = x^2 + 2xy + x {/eq}

Therefore:

{eq}\frac{dY}{dX} = -\frac{F_{x}}{F_{y}} = \displaystyle -\frac{2xy + y^2 + y}{x^2 + 2xy + x} {/eq}