How do you find dy/dx by implicit differentiation for the equation x² + xy + y² = 9?

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To find
dy/dx
by implicit differentiation for the equation:

x² + xy + y² = 9

we will differentiate both sides of the equation with respect to x. Remember, when differentiating terms that include y, we need to apply the chain rule since y is a function of x.

Step 1: Differentiate each term

  • Differentiate x²:
    d/dx (x²) = 2x
  • Differentiate xy:
    Here, we use the product rule. Let:
    u = x and v = y, so:
    d/dx (xy) = u (dv/dx) + v (du/dx) = y + x(dy/dx)
  • Differentiate y²:
    Using the chain rule:
    d/dx (y²) = 2y(dy/dx)

Step 2: Differentiate the right side

The right side is a constant (9), so its derivative is:

  • d/dx (9) = 0

Step 3: Combine all the derivatives

Now, putting everything together, we get:

2x + (y + x(dy/dx)) + 2y(dy/dx) = 0

Step 4: Solve for dy/dx

Now we need to isolate dy/dx:

x(dy/dx) + 2y(dy/dx) = -2x - y

Factor out dy/dx:

dy/dx (x + 2y) = -2x - y

Finally, divide by (x + 2y) to solve for dy/dx:

dy/dx = (-2x - y) / (x + 2y)

Result

Thus, the derivative dy/dx for the implicit equation x² + xy + y² = 9 is:

dy/dx = (-2x - y) / (x + 2y)

And that’s how you find the derivative using implicit differentiation!

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