Single Variable Calculus, Chapter 3, 3.2, Section 3.2, Problem 28

a.) Illustrate the graph of $f(x) = \sqrt{6 - x}$







b.) Sketch the graph of $f'$ using the graph from part (a)







c.) Find $f'(x)$ using the definition of a derivative. State the domains of $f$ and $f'$.

Using the definition of derivative


$
\begin{equation}
\begin{aligned}

\qquad f'(x) =& \lim_{h \to 0} \frac{f(t + h) = f(t)}{h}
&&
\\
\\
\qquad f'(x) =& \lim_{h \to 0} \frac{\sqrt{6 - (x + h)} - \sqrt{ 6 -x}}{h}
&& \text{Substitute $f(x + h)$ and $f(x)$}
\\
\\
\qquad f'(x) =& \lim_{h \to 0} \frac{\sqrt{6 - x - h} - \sqrt{ 6 -x}}{h} \cdot \frac{\sqrt{6 - x - h} + \sqrt{ 6 -x}}{\sqrt{6 - x - h} + \sqrt{ 6 -x}}
&& \text{Multiply both numerator and denominator by $(\sqrt{6 - x - h} + \sqrt{ 6 -x})$}
\\
\\
\qquad f'(x) =& \lim_{h \to 0} \frac{6 - x - h - \cancel{\sqrt{(6 - x - h) + (6 -x)}} + \cancel{\sqrt{(6 - x - h) + (6 -x)}} - (6 -x )}{h(\sqrt{6 - x - h} + \sqrt{ 6 -x})}
&& \text{Combine like terms}
\\
\\
\qquad f'(x) =& \lim_{h \to 0} \frac{\cancel{6} - \cancel{x} - h - \cancel{6} + \cancel{x}}{h(\sqrt{6 - x - h} + \sqrt{ 6 -x})}
&& \text{}
\\
\\
\qquad f'(x) =& \lim_{h \to 0} \frac{-\cancel{h}}{\cancel{h}(\sqrt{6 - x - h} + \sqrt{ 6 -x})}
&& \text{Cancel out like terms}
\\
\\
f'(x) =& \lim_{h \to 0} \left( \frac{-1}{\sqrt{6 - x - h} + \sqrt{6 - x}} \right) = \frac{-1}{\sqrt{6 - x - 0} + \sqrt{6 - x}} = \frac{-1}{\sqrt{6 - x} + \sqrt{ 6 - x}}
&& \text{Evaluate the limit}
&&
\\
\\
f'(x) =& \frac{-1}{2 \sqrt{6 -x}}
&&

\end{aligned}
\end{equation}
$


Both $f(x)$ and $f'(x)$ are root functions that are continuous for every positive values of $x$. However, the square root is placed in the denominator of $f'(x)$ making the function defined only for $6 - x > 0$.

$
\begin{array}{cc}
\text{For } f(x) \qquad & \text{For } f'(x) \\
6 - x \geq 0 \qquad & 6 - x > 0 \\
\, \, \, \, \, x \leq 6 \qquad & \, \, \, \, \, x < 6
\end{array}
$


Therefore, the domain of $f(x)$ is $(-\infty, 6]$ while the domain of $f'(x)$ is $(-\infty, 6)$

d.) Graph $f'$ and compare with your sketch in part (b)