Differentiate the following by delta method :
`(x-1)(x-2)`

Answer

To differentiate the function \( y = (x-1)(x-2) \) using the delta method, we will follow these steps:

### Step 1: Define the function
Let:
\[ y = f(x) = (x-1)(x-2) \]

### Step 2: Calculate \( f(x + h) \)
We need to find \( f(x + h) \):
\[ f(x + h) = (x + h - 1)(x + h - 2) \]
Simplifying this:
\[ = (x + h - 1)(x + h - 2) = (x + h - 1)(x + h - 2) = (x + h - 1)(x + h - 2) \]
\[ = (x + h - 1)(x + h - 2) = (x + h - 1)(x + h - 2) = (x + h - 1)(x + h - 2) \]
Expanding this:
\[ = (x + h - 1)(x + h - 2) = (x + h - 1)(x + h - 2) = (x + h - 1)(x + h - 2) \]
\[ = (x^2 + hx - 2x + hx + h^2 - 2h - x + 2) = x^2 + 2hx + h^2 - 3x + 2 - 2h \]

### Step 3: Set up the difference quotient
Now, we can find \( f(x + h) - f(x) \):
\[ f(x + h) - f(x) = [x^2 + 2hx + h^2 - 3x + 2 - 2h] - [(x-1)(x-2)] \]
Calculating \( f(x) \):
\[ f(x) = x^2 - 3x + 2 \]
Thus:
\[ f(x + h) - f(x) = [x^2 + 2hx + h^2 - 3x + 2 - 2h] - [x^2 - 3x + 2] \]
\[ = 2hx + h^2 - 2h \]

### Step 4: Divide by \( h \)
Now, we divide by \( h \):
\[ \frac{f(x + h) - f(x)}{h} = \frac{2hx + h^2 - 2h}{h} = 2x + h - 2 \]

### Step 5: Take the limit as \( h \to 0 \)
Now we take the limit as \( h \) approaches 0:
\[ \lim_{h \to 0} \left( 2x + h - 2 \right) = 2x - 2 \]

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 2x - 3 \]