Find `dy/dx if x= y^2-6y+11`

Answer

To find \(\frac{dy}{dx}\) given the equation \(x = y^2 - 6y + 11\), we will use implicit differentiation. Here are the steps to solve the problem:

### Step 1: Differentiate both sides with respect to \(x\)
We start with the equation:
\[ x = y^2 - 6y + 11 \]
Differentiating both sides with respect to \(x\):
\[ \frac{dx}{dx} = \frac{d}{dx}(y^2 - 6y + 11) \]
This simplifies to:
\[ 1 = \frac{d}{dx}(y^2) - \frac{d}{dx}(6y) + \frac{d}{dx}(11) \]

### Step 2: Apply the differentiation rules
Using the chain rule for differentiation:
- The derivative of \(y^2\) is \(2y \cdot \frac{dy}{dx}\)
- The derivative of \(6y\) is \(6 \cdot \frac{dy}{dx}\)
- The derivative of a constant (11) is 0

So we have:
\[ 1 = 2y \cdot \frac{dy}{dx} - 6 \cdot \frac{dy}{dx} + 0 \]

### Step 3: Combine like terms
Now we can combine the terms involving \(\frac{dy}{dx}\):
\[ 1 = (2y - 6) \cdot \frac{dy}{dx} \]

### Step 4: Solve for \(\frac{dy}{dx}\)
To isolate \(\frac{dy}{dx}\), we divide both sides by \((2y - 6)\):
\[ \frac{dy}{dx} = \frac{1}{2y - 6} \]

### Final Result
Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{1}{2y - 6} \]