To find the slope of the function \( y = x - x^{2} \) at \( x = 1 \), we need to calculate the derivative of the function and evaluate it at the given point.

Slope Calculation Using Derivative

The slope of a function at any point is given by its first derivative, \( \frac{dy}{dx} \). First, find the derivative of the function \( y = x - x^{2} \):

Using the power rule for differentiation ( \(\frac{d}{dx}(x^n) = nx^{n-1}\) ): $ \frac{dy}{dx} = \frac{d}{dx}(x) - \frac{d}{dx}(x^{2}) $ $ \frac{dy}{dx} = 1 \cdot x^{1-1} - 2 \cdot x^{2-1} $ $ \frac{dy}{dx} = 1 \cdot x^{0} - 2 \cdot x^{1} $ $ \frac{dy}{dx} = 1 - 2x $

Evaluating Derivative at x = 1

Now, substitute \( x = 1 \) into the derivative to find the slope at that specific point:

$ \frac{dy}{dx} \Big|_{x=1} = 1 - 2(1) $ $ \frac{dy}{dx} \Big|_{x=1} = 1 - 2 $ $ \frac{dy}{dx} \Big|_{x=1} = -1 $

The slope of the function \( y = x - x^{2} \) at \( x = 1 \) is -1.