SOLUTION: Let y be defined implicitly by the equation ln(5y)=3xy. Use implicit differentiation to find the second derivative of y with respect to x. d^2y/dx^2 = Simplify fully please!

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Question 1200776: Let y be defined implicitly by the equation ln(5y)=3xy. Use implicit differentiation to find the second derivative of y with respect to x.
d^2y/dx^2 =
Simplify fully please!
Answer by asinus(45) (Show Source):
You can put this solution on YOUR website!
**1. Differentiate Implicitly**
* **Original Equation:**
ln(5y) = 3xy
* **Differentiate both sides with respect to x:**
(1 / (5y)) * (5 * dy/dx) = 3y + 3x * dy/dx
* **Simplify:**
(1 / y) * dy/dx = 3y + 3x * dy/dx
* **Isolate dy/dx:**
(1 / y) * dy/dx - 3x * dy/dx = 3y
dy/dx * (1/y - 3x) = 3y
dy/dx = (3y) / (1/y - 3x)
dy/dx = (3y²) / (1 - 3xy)
**2. Differentiate Implicitly Again**
* **Differentiate both sides of dy/dx with respect to x:**
d²y/dx² = [(3 * 2y * dy/dx) * (1 - 3xy) - (3y²) * (-3y)] / (1 - 3xy)²
* **Substitute the expression for dy/dx:**
d²y/dx² = [(6y * (3y²) / (1 - 3xy)) * (1 - 3xy) - (3y²) * (-3y)] / (1 - 3xy)²
* **Simplify:**
d²y/dx² = [18y³ - (-9y³)] / (1 - 3xy)²
d²y/dx² = (27y³) / (1 - 3xy)²
**Therefore:**
**d²y/dx² = (27y³) / (1 - 3xy)²**
This is the second derivative of y with respect to x.