`4x^3 + ln(y^2) + 2y = 2x` Use implicit differentiation to find dy/dx

Find `(dy)/(dx) ` if `4x^3+lny^2+2y=2x `

Rewrite the second term using a property of logarithms:

`4x^3+2lny+2y=2x `

Divide through by 2:

`2x^3+lny+y=x `

Differentiate term by term with respect to x:

`6x^2+1/y*(dy)/(dx)+(dy)/(dx)=1 `

`(dy)/(dx)(1/y+1)=1-6x^2 `

`(dy)/(dx)=(1-6x^2)/(1/y+y) `

`(dy)/(dx)=(y-6x^2y)/(1+y^2) `

Find `(dy)/(dx) ` if `4x^3+lny^2+2y=2x `

Rewrite the second term using a property of logarithms:

`4x^3+2lny+2y=2x `

Divide through by 2:

`2x^3+lny+y=x `

Differentiate term by term with respect to x:

`6x^2+1/y*(dy)/(dx)+(dy)/(dx)=1 `

`(dy)/(dx)(1/y+1)=1-6x^2 `

`(dy)/(dx)=(1-6x^2)/(1/y+y) `

`(dy)/(dx)=(y-6x^2y)/(1+y^2) `