Q. What is the slope of the function $$y = x + \frac{1}{x}$$ at $x = 0.5$?
Soln. Simply differentiate $y$ with respect to $x$, to get $$\frac{dy}{dx} = 1 - \frac{1}{x^2}$$
We also know that the slope of $y$ at $x_0$ is given by $\left(\frac{dy}{dx}\right)_{y_0}$
$$\Rightarrow slope = \left(\frac{dy}{dx}\right)_{x_0 = 0.5}$$ $$\Rightarrow \left(\frac{dy}{dx}\right)_{x_0 = 0.5} = 1 - \frac{1}{0.5^2}$$ $$\Rightarrow slope = 1 - \frac{1}{0.25}$$ $$\Rightarrow slope = -3$$
Thus the required slope of the function $y = x + \frac{1}{x}$ at $x = 0.5$ is $-3$.
Soln. Simply differentiate $y$ with respect to $x$, to get $$\frac{dy}{dx} = 1 - \frac{1}{x^2}$$
We also know that the slope of $y$ at $x_0$ is given by $\left(\frac{dy}{dx}\right)_{y_0}$
$$\Rightarrow slope = \left(\frac{dy}{dx}\right)_{x_0 = 0.5}$$ $$\Rightarrow \left(\frac{dy}{dx}\right)_{x_0 = 0.5} = 1 - \frac{1}{0.5^2}$$ $$\Rightarrow slope = 1 - \frac{1}{0.25}$$ $$\Rightarrow slope = -3$$
Thus the required slope of the function $y = x + \frac{1}{x}$ at $x = 0.5$ is $-3$.
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