Use implicit differentiation to find $$ \frac{d y}{d x} $$. $$ 9 x y+y^{2}=2 x+y $$

Question

UpStudy AI · Accepted Answer

We start with the equation

9xy + y² = 2x + y.

To find dy/dx using implicit differentiation, differentiate both sides of the equation with respect to x.

Step 1. Differentiate each term:

• For 9xy, use the product rule (derivative of x·y is y + x·(dy/dx)):
  d/dx[9xy] = 9(x·(dy/dx) + y).

• For y², use the chain rule:
  d/dx[y²] = 2y · (dy/dx).

• For 2x, the derivative is:
  d/dx[2x] = 2.

• For y, the derivative is:
  d/dx[y] = dy/dx.

Step 2. Write the differentiated equation:

9(x·(dy/dx) + y) + 2y·(dy/dx) = 2 + (dy/dx).

Step 3. Expand and gather like terms:

9x·(dy/dx) + 9y + 2y·(dy/dx) = 2 + (dy/dx).

Bring all terms involving dy/dx to one side by subtracting (dy/dx) from both sides:

9x·(dy/dx) + 2y·(dy/dx) − (dy/dx) = 2 − 9y.

Factor out dy/dx from the left side:

(9x + 2y − 1)·(dy/dx) = 2 − 9y.

Step 4. Solve for dy/dx:

dy/dx = (2 − 9y) / (9x + 2y − 1).

Thus, the derivative dy/dx is given by

dy/dx = (2 − 9y) / (9x + 2y − 1).
$$ \frac{dy}{dx} = \frac{2 - 9y}{9x + 2y - 1} $$

Use implicit differentiation to find \( \frac{d y}{d x} \). \[ 9 x y+y^{2}=2 x+y \]