# Single Variable Calculus, Chapter 3, 3.9, Section 3.9, Problem 20

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Calculate $\delta y$ and $dy$ of $y = \sqrt{x}$ for $x = 1$ and $dx = \delta x = 1$. Then sketch a diagram showing the line segments with lengths $dx, dy$ and $\delta y$

Solving for $\delta y$

$\delta y = f(x + \delta x) - f(x)$

\begin{aligned} f(x + \delta x) =& f(1 + 1) = f(2) = \sqrt{2} \\ \\ f(x) =& f(1) = \sqrt{1} \\ \\ f(1) =& 1 \\ \\ \delta y =& f(2) - f(1) \\ \\ \delta y =& \sqrt{2} - 1 \\ \\ \delta y =& 0.41 \end{aligned}

Solving for $dy$

\begin{aligned} dy =& f'(x) dx \\ \\ \frac{dy}{dx} =& \frac{d}{dx} (x)^{\frac{1}{2}} \\ \\ \frac{dy}{dx} =& \frac{1}{2} (x)^{\frac{-1}{2}} \\ \\ dy =& \frac{1}{2 \sqrt{x}} dx \\ \\ dy =& \left( \frac{1}{2 \sqrt{1}} \right) (1) \\ \\ dy =& \frac{1}{2(1)} \\ \\ dy =& \frac{1}{2} \end{aligned}