Implicit differentiation calculus question!!?

Question

Use the process of implicit differentiation to find dy/dx given that, 3x^4y^5 + 5y^3/6x^2 = 7xy

Answer 1

The equation is 3x⁴ y⁵ + (5/6)y³/x² = 7xy.

Rewrite the equation as 3x⁴ y⁵ + (5/6)y³x^{- 2} = 7xy.

Differentiate the above equation with respect to x.

Use product rule : (uv) ' = u ' v + v ' u.

d/dx[3x⁴ y⁵ + (5/6)y³x^{- 2} ] = d/dx[7xy]

3(x⁴ y⁵) ' + (5/6)(y³x^{- 2}) ' = 7(xy) '

3(x⁴)' y⁵ + 3(y⁵)' x⁴ + (5/6)(y³)' x^{- 2} + (5/6)(x^{- 2} )' y³ = 7xy' + 7yx'

Use power rule : (xⁿ)' =nx^{n - 1} .

12x³y⁵ + 15y⁴x⁴y'+ (5/2)y²x^{- 2}y' - (5/3)x^{- 3}y³ - 7xy' = 7y

y'[ 15y⁴x⁴ + (5/2)y²x^{- 2} - 7x ] = 7y - 12x³y⁵ + (5/3)x^{- 3}y³

⇒ y' = [ 7y - 12x³y⁵ + (5/3)x^{- 3}y³ ]/[ 15y⁴x⁴ + (5/2)y²x^{- 2} - 7x ].

Therefore, dy/dx = [ 7y - 12x³y⁵ + (5/3)x^{- 3}y³ ]/[ 15y⁴x⁴ + (5/2)y²x^{- 2} - 7x ].