# please explain how to use implicit differentiation to obtain: `dy/dx` `[x^2y+5xy^3-x=2]`

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### 1 Answer

The given equation is: `x^2y+5xy^3-x=2`

y is not an explicit function of x, hence in order to find `dy/dx` , recourse of implicit differentiation has to be resorted to.

Differentiate both sides of the equation with respect to x, getting

`d/dx(x^2y)+5d/dx(xy^3)-d/dx(x)=d/dx(2)`

Use the product rule twice, and note that `d/dx(phi(y))=d/dy(phi(y)) * dy/dx`

`rArr d/dx(x^2)*y+d/dx(y)*x^2+5[d/dx(x)*y^3+d/dx(y^3)*x]-d/dx(x)=d/dx(2)`

`rArr 2xy+x^2*dy/dx+5y^3*1+5x*3y^2dy/dx-1=0`

`rArr dy/dx(x^2+15xy^2)=1-2xy-5y^3`

`rArr dy/dx=(1-2xy-5y^3)/(x^2+15xy^2)`

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