Determine dy/dx using implicit differentiation for y^2*x + 3y*x^2 = 8

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The derivative `dy/dx` has to be determined given that y^2*x + 3y*x^2 = 8. Use implicit differentiation and differentiate both sides of the equation with respect to x.

`(d(y^2*x + 3y*x^2))/(dx) = (d(8))/dx`

=> `(d(y^2*x))/dx + (d(3y*x^2))/dx = 0`

=> `2y*y'*x + y^2 + 3x^2*y' + 3y*2x = 0`

=> `y'(2y*x + 3x^2) + y^2 + 6yx = 0`

=> `y'(2y*x + 3x^2) = -(y^2 + 6yx)`

=> `y' = -(y^2 + 6yx)/(2y*x + 3x^2)`

For y^2*x + 3y*x^2 = 8, the derivative

`dy/dx = -(y^2 + 6yx)/(2y*x + 3x^2) `

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