Question

Use implicit differentiation to find {d y}{d x} [ (6 x y+5)² =36 y

Answer 1

To find dy/dx using implicit differentiation for the equation (6xy + 5)² = 36y, differentiate both sides with respect to x, apply the chain and product rules, and solve for dy/dx.

Step-by-step explanation:

To use implicit differentiation to find dy/dx for the equation (6xy + 5)² = 36y, follow these steps:

1. Differentiate both sides of the equation with respect to x using the chain rule for the left side and the constant rule and power rule for the right side.
2. For the left side, (d/dx)(6xy + 5)² = 2(6xy + 5) ⋅ (d/dx)(6xy + 5).
3. Now apply the product rule for (d/dx)(6xy): d/dx(6xy) = 6(y + x(dy/dx)).
4. For the right side, differentiate 36y with respect to x to get 36(dy/dx).
5. Combine the derivatives and solve for dy/dx to get the final result.

Following these steps will give you the derivative dy/dx for the given equation.