Find the implicited derivative

Question

dy/dx=5y^2/1-5xy

Find the implicit differentiation to find the second derivative of y with the respect to x.

d^2y/dx^2 = ?

Answer 1

dy/dx = y' = 5y²/(1 – 5xy)

Differentiate using the quotient rule

d²y/dx² = y'' = [(1-5xy)10yy' + 5y²(5y + 5xy')]/(1 – 5xy)²

Substitute y' = [5y²/(1 – 5xy)]

y'' = [(1 – 5xy)10y[5y²/(1 – 5xy)] + 5y²(5y + 5x[5y²/(1 – 5xy)])]/(1 – 5xy)²

Multiply numerator and denominator by (1 – 5xy)

y'' = [5y²(1 – 5xy)10y + 5y²(5y(1 – 5xy) + 5x(5y²))]/(1 – 5xy)³

Use the common factor 5y² to factor the numerator

y'' = 5y²[10y(1 – 5xy) + 5y(1 – 5xy) + 25xy²)]/(1 – 5xy)³

Further simplifications give us

y'' = 5y²[(15y(1 – 5xy) + 25xy²)]/(1 – 5xy)³ = 5y²(15y – 75xy² + 25xy²)/(1 – 5xy)³ = 5y²(15y – 50xy²)/(1 – 5xy)³

Factor the common 5y factor out of the parenthesis in the numerator to give

25y³(3 – 10xy)/(1 – 5xy)³

Answer

d²y/dx² = 25y³(3 – 10xy)/(1 – 5xy)³

Answer 2

dy/dx = 5y² / (1 - 5xy)

d²y / dx² = [(1 - 5xy)(10y)(dy/dx) - 5y²(-5y - 5x(dy/dx))] / (1 - 5xy)²

= [50y³ + 25y³+ 125xy⁴ / (1 - 5xy)] / (1 - 5xy)²

= [75y³(1 - 5xy) + 125xy⁴] / (1 - 5xy)³ = (75y³ - 250xy⁴) / (1 - 5xy)³