To find dy/dx using implicit differentiation for the equation x² + 6xy + y² = 6, we will differentiate both sides of the equation with respect to x.
1. **Differentiate the left side:**
- When differentiating x², we get 2x.
- For 6xy, we apply the product rule: the derivative is 6(y + x(dy/dx)).
- For y², using the chain rule, we get 2y(dy/dx).
So, the left-hand side becomes:
2x + 6(y + x(dy/dx)) + 2y(dy/dx)
2. **Differentiate the right side:**
- The right side, 6, differentiates to 0.
Putting it all together:
2x + 6(y + x(dy/dx)) + 2y(dy/dx) = 0
3. **Rearranging to solve for dy/dx:**
We will group the terms containing dy/dx:
6x(dy/dx) + 2y(dy/dx) = -2x – 6y
4. **Factor out dy/dx:**
dy/dx(6x + 2y) = -2x – 6y
5. **Solve for dy/dx:**
dy/dx = (-2x – 6y) / (6x + 2y)
This equation represents the derivative of y with respect to x based on the initial equation. In summary, we used implicit differentiation to differentiate both sides and solve for dy/dx, yielding the final result.