Question

Find dy/dx by implicit differentiation.
x^4(x+y)=y^2(3x-y)

Answer 1

x^4 (x + y) = y^2 (3x - y)

expand

x^5 - (x^4) (y) = 3(x)(y^2) - y^3

now differentiate implicitly

5x^4 - x^4 (y ' ) - (y)(4x^3) = (3x)(2y y' ) - (3y^2)(1) - 3(y^2)(y ' )

=> 5x^4 - x^4 (y ' ) - 4(y)(x^3) = 6(x)(y) (y' ) - (3y^2) - 3(y^2)( y ' )

=> y ' [ 6xy + x^4 - 3y^2 ] = 5x^4 - 4 x^3 y + 3y^2

y ' = [ 5x^4 - 4 x^3 y + 3y^2 ] / [6xy + x^4 - 3y^2 ]

Answer 2

thx for the medal