how to find the derivatives of implicit function xy^2 + (xy)^1/3 + x^4 = 7 with respect to x?

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jeew-m eNotes educator| Certified Educator

`xy^2 + (xy)^1/3 + x^4 = 7`


Here we need to know the derivative of function of a function. Lets consider a function g(y). lets say it is a function of y . Then y is a function of x. So when we differentiate according x we need to do it as follows.

First we need to differentiate g(y) with respect to y and then y with respect to x.

`(d(g(y)))/dx = (d(g(y)))/dy*(dy)/dx`


If you consider `g(y) = y^2` then;

`(d(y^2))/dx `

`= (d(y^2))/dy*(dy)/dx`

`= 2y*(dy)/dx`


`xy^2 + (xy)^1/3 + x^4 = 7`

`x*2y(dy/dx)+y^2+1/3(xy)^(1/3-1)[x*(dy)/dx+y]+4x^3 = 0`

`(dy)/dx[2yx+1/3*(xy)^(-2/3)x]+y^2+1/3*(xy)^(-2/3)y+4x^3 = 0`


`(dy)/dx[2yx+1/3*(xy)^(-2/3)x] = -[y^2+4x^3+1/3*(xy)^(-2/3)y]`


`(dy)/dx = -[y^2+4x^3+1/3*(xy)^(-2/3)y]/[x(2y+1/3*(xy)^(-2/3))]`