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

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{equation}
\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}
\end{equation}
$


Solving for $dy$


$
\begin{equation}
\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}
\end{equation}
$