For the given function, (a) find the slope of the tangent line to the graph...

Step 1
The function is given b y
${\displaystyle h\left(x\right)=\frac{7}{x}atx=6}$ To evaluate:
(a) The slope of the tangent line to the graph of the function at ${\displaystyle x=6}$
(b) The equation of the tangent line to the graph of the function at ${\displaystyle x=6}$
(a) Evaluation of the slope The slope of the tangent line to the graph of the function can be evaluat6ed as following
${\displaystyle m=\frac{dh}{dx}{\mid }_{x=6}}$
${\displaystyle =\frac{d}{dx}\left(\frac{7}{x}\right){\mid }_{x=6}}$
${\displaystyle =-\frac{7}{x^2}{\mid }_{x=6}}$
${\displaystyle =-\frac{7}{6^2}}$
$=-\frac{7}{36}$
Hence, the slope of the tangent line at ${\displaystyle x=6is-736}$

(b) Evaluation of the equation of the tangent line
The equation of the tangent line can be evaluated as following
${\displaystyle y-h\left(6\right)=m(x-6)}$
${\displaystyle \Rightarrow y-\frac{7}{6}=-\frac{7}{36}(x-6)}$
${\displaystyle \Rightarrow y-\frac{7}{6}=-\frac{7}{3x+\frac{7}{6}}}$
$\Rightarrow y=-\frac{7}{36}x+\frac{7}{6}+\frac{7}{6}$
${\displaystyle \Rightarrow y=-\frac{7}{36}x+\frac{7}{3}}$
Hence, the equation of the tangent line at ${\displaystyle x=6}$ is
${\displaystyle y=-\frac{7}{36}x+\frac{7}{3}}$