Use implicit differentiation to find \( \frac{d y}{d x} \) \( 5 y^{2}=\frac{3 x-2}{3 x+2} \)

We start with the equation   5y² = (3x – 2)/(3x + 2). Step 1. Differentiate both sides with respect to x. Remember that y is a function of x, so when differentiating y², use the chain rule. The derivative of the left-hand side (LHS) is:   d/dx(5y²) = 5·2y·(dy/dx) = 10y·(dy/dx). Step 2. Differentiate the right-hand side (RHS). We have a quotient, so use the quotient rule. For a function f(x)/g(x),   d/dx[f(x)/g(x)] = (f′(x)·g(x) – f(x)·g′(x))/(g(x))². Let f(x) = 3x – 2 and g(x) = 3x + 2. Then:   f′(x) = 3  and  g′(x) = 3. Thus,   d/dx[(3x – 2)/(3x + 2)] = [3·(3x + 2) – (3x – 2)·3] / (3x + 2)². Simplify the numerator:   3(3x + 2) – 3(3x – 2) = (9x + 6) – (9x – 6) = 12. So the derivative of the RHS is:   12 / (3x + 2)². Step 3. Set the derivatives equal to each other:   10y·(dy/dx) = 12/(3x + 2)². Step 4. Solve for dy/dx by isolating it:   dy/dx = 12 / [10y·(3x + 2)²]. Simplify the fraction (divide numerator and denominator by 2):   dy/dx = 6 / [5y·(3x + 2)²]. Thus, the derivative of y with respect to x is:   dy/dx = 6/(5y(3x + 2)²).