Question 1200776
**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.