The slope of the function \( y = x - x^{2} \) at \( x = 1 \) is ____________
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.
Important Questions from Calculus
The ratio of volume to surface area of solid semi sphere is related to its radius through
The value of \(\int^2_0\int^x_0y\ dy\ dx\)
Find the slope of normal to the curve y = x2 + 7x at (1, 8).
Find the equation of normal to the curve y = 4x - 3x2 at (2, -4).
- \(\mathop {\lim }\limits_{\theta \to \frac{\pi }{2}} \frac{{\log \left( {\theta - \frac{\pi }{2}} \right)}}{{\tan \theta }}\)
