Consider the equation `x+|y|=2y`
What is the derivative of y as a function of x with respect x for `xlt0` ?

Options

A. 2
B. 1
C. `1/2`
D. `1/3`

Answer

To find the derivative of \( y \) as a function of \( x \) with respect to \( x \) for \( x < 0 \) given the equation \( x + |y| = 2y \), we can follow these steps:

### Step 1: Analyze the equation
The equation given is:
\[ x + |y| = 2y \]
For \( x < 0 \), we need to consider the cases for \( |y| \). The absolute value function \( |y| \) can be expressed as:
- \( |y| = y \) if \( y \geq 0 \)
- \( |y| = -y \) if \( y < 0 \)

### Step 2: Solve for \( y \) when \( y < 0 \)
Since we are interested in the case where \( x < 0 \), we will assume \( y < 0 \). Thus, we replace \( |y| \) with \( -y \):
\[ x - y = 2y \]
Rearranging this gives:
\[ x = 2y + y \]
\[ x = 3y \]
Now, solving for \( y \):
\[ y = \frac{x}{3} \]

### Step 3: Differentiate \( y \) with respect to \( x \)
Now that we have \( y \) as a function of \( x \):
\[ y = \frac{x}{3} \]
We can differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{3} \]

### Conclusion
Thus, the derivative of \( y \) with respect to \( x \) for \( x < 0 \) is:
\[ \frac{dy}{dx} = \frac{1}{3} \]