Find the slope of tangent line to the curve :
`y=x^(2)-2x+1.`

Answer

To find the slope of the tangent line to the curve given by the equation \( y = x^2 - 2x + 1 \), we need to follow these steps:

### Step 1: Differentiate the function
We start by differentiating the function \( y \) with respect to \( x \). The derivative \( \frac{dy}{dx} \) gives us the slope of the tangent line at any point on the curve.

Given:
\[ y = x^2 - 2x + 1 \]

Using the power rule for differentiation:
- The derivative of \( x^2 \) is \( 2x \).
- The derivative of \( -2x \) is \( -2 \).
- The derivative of a constant (1) is \( 0 \).

So, we differentiate:
\[ \frac{dy}{dx} = 2x - 2 \]

### Step 2: Interpret the result
The expression \( \frac{dy}{dx} = 2x - 2 \) represents the slope of the tangent line to the curve at any point \( x \).

### Step 3: Find the slope at a specific point (if needed)
If we want to find the slope of the tangent line at a specific point, we can substitute that \( x \) value into the derivative. For example, if we want to find the slope at \( x = 1 \):
\[ \frac{dy}{dx} \bigg|_{x=1} = 2(1) - 2 = 0 \]

### Final Answer
The slope of the tangent line to the curve \( y = x^2 - 2x + 1 \) is given by the expression \( 2x - 2 \). At \( x = 1 \), the slope is \( 0 \).

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