Implicit differentiation is basically done in cases where yy cannot be explicitly written as a function of xx.
In this case,
-3=(x^2+y)^3-y^2x −3=(x2+y)3−y2x
Differentiating both sides w.r.t. xx
=> -3(d(1))/dx = [d(x^2+y)^3]/dx - (d(y^2x))/dx⇒−3d(1)dx=d(x2+y)3dx−d(y2x)dx
**Using chain rule to evaluate [d(x^2+y)^3]/dxd(x2+y)3dx & product rule to evaluate (d(y^2x))/dxd(y2x)dx **
=> 0 = [d(x^2+y)^3]/(d(x^2+y))*[d(x^2+y)]/dx - [y^2dx/dx + xdy^2/dx]⇒0=d(x2+y)3d(x2+y)⋅d(x2+y)dx−[y2dxdx+xdy2dx]
=> y^2dx/dx + xdy^2/dx = [d(x^2+y)^3]/(d(x^2+y))*[d(x^2+y)]/dx⇒y2dxdx+xdy2dx=d(x2+y)3d(x2+y)⋅d(x2+y)dx
Using sum rule to evaluate [d(x^2+y)]/dxd(x2+y)dx & chain rule to evaluate dy^2/dxdy2dx
=> y^2 + x*dy^2/dy*dy/dx = [3*(x^2+y)^2]*[dx^2/dx + dy/dx]⇒y2+x⋅dy2dy⋅dydx=[3⋅(x2+y)2]⋅[dx2dx+dydx]
=> y^2 + x*2y*dy/dx = 3(x^2+y)^2*[2x+dy/dx]⇒y2+x⋅2y⋅dydx=3(x2+y)2⋅[2x+dydx]
=> y^2 + 2xy*dy/dx = 6x(x^2+y)^2 + 3(x^2+y)^2*dy/dx⇒y2+2xy⋅dydx=6x(x2+y)2+3(x2+y)2⋅dydx
=> 2xy*dy/dx - 3(x^2+y)^2*dy/dx = 6x(x^2+y)^2 - y^2 ⇒2xy⋅dydx−3(x2+y)2⋅dydx=6x(x2+y)2−y2
=> [2xy - 3(x^2+y)^2]*dy/dx = 6x(x^2+y)^2 - y^2⇒[2xy−3(x2+y)2]⋅dydx=6x(x2+y)2−y2
=> color(red){dy/dx = [6x(x^2+y)^2 - y^2]/[2xy - 3(x^2+y)^2]}⇒dydx=6x(x2+y)2−y22xy−3(x2+y)2
