Answer
$-\displaystyle \frac{1}{2}$
Work Step by Step
The slope of the tangent line through the point on the graph of $f$ where $x=a$
is given by the instantaneous rate of change, or derivative
$m_{tan}=$ slope of tangent $=$ instantaneous rate of change$=$derivative
$=f^{\prime}(a)=\displaystyle \lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h} ,$
assuming the limit exists.
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Let P be $(a,f(a))$, since it is on the graph of f.
From the graph, we read (estimate) two points from the tangent line at P:
(4,2) and (6,1)
(we estimate the line as being horizontal )
The slope of the tangent line :
$m=\displaystyle \frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{1-2}{6-4}=\frac{-1}{2}= -\displaystyle \frac{1}{2}$
so, we estimate
$f^{\prime}(a)=m=-\displaystyle \frac{1}{2}$
