Answer
$-5.5$
Work Step by Step
The derivative, or instantaneous rate of change of $f(x)$ at $x=a$ is defined as
$f^{\prime}(a)=\displaystyle \lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h},\ \ \ \ $(*) if the limit exists.
(A limit at x=a exists if both one-sided limits exist, and
the one-sided limits are equal.)
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The first two rows indicate that the right sided limit exists, because
as h approaches zero from the right (the $+$ side),
$\displaystyle \frac{r(a+h)-r(a)}{h}$ approaches $-5.5$.
The last two rows indicate that the left sided limit exists, because
as h approaches zero from the left (the - side),
$\displaystyle \frac{r(a+h)-r(a)}{h}$ approaches $-5.5$.
Since we estimate that both one-sided limits exist and equal $-5.5$,
the limit (*) exists and equals $-5.5$, so
our estimate for $r^{\prime}(-6)$ is
$r^{\prime}(-6)=-5.5$
