Find dy/dx if x^5 + 4xy^3 – y^5 =2

To find dy/dx if x^5 + 4xy^3 – y^5 =
2.

The relation between x and y in this equation is
implicit. In such cases we straight away differentiate term by terrm and  try to get
dy/dx by solving for dy/dx  from the
equation.

Differentiating both sides of the given equation
with respect to x, we get:

(x^5 + 4xy^3 – y^5)'
=(2)'

(x^5)' + {4xy^3}' – {y^5}' =
0

{x

5x^4 +{4(x)'y^3
+4x(y^3)'}-{5y^4*dy/dx} = 0.

5x^4+4y^3
+4x*3y^2dy/dx-5y^4dy/dx = 0. We try to solve for dy/dx from this
equation:

5x^x+4y^3 +{12x^3y^2 - 5y^4) dy/dx =
0

(12x^3y^2-5y^4)dy/dx.

Divide
both sides by the coefficient of dy/dx, that is,
12x^2-5y^4.

dy/dx = -(5x^4+4y^3)/(12x^3y^2-5y^4).
Or

dy/dx =
(4y^3+5x^4)/(5y^4-12x^3y^4)