Answer
$m=12$
Work Step by Step
$f(x)=x^{3}+1,$ at $(2,9)$
The slope of the tangent line at the point $P(a,f(a))$ is given by $m=\lim_{h\to0}\dfrac{f(a+h)-f(a)}{h}$
In this case, $a=2$
Find $f(a+h)$ by substituting $x$ by $2+h$ in $f(x)$ and simplifying:
$f(2+h)=(2+h)^{3}+1=2^{3}+3(2)^{2}h+3(2)h^{2}+h^{3}+1=...$
$...=8+12h+6h^{2}+h^{3}+1=h^{3}+6h^{2}+12h+9$
Since $f(a)$ is the $y$-coordinate of the point given, $f(a)=9$
Substitute the known values into the formula that gives the slope of the tangent line and evaluate:
$m=\lim_{h\to0}\dfrac{h^{3}+6h^{2}+12h+9-9}{h}=...$
$...=\lim_{h\to0}\dfrac{h^{3}+6h^{2}+12h}{h}=\lim_{h\to0}\dfrac{h(h^{2}+6h+12)}{h}=...$
$...=\lim_{h\to0}(h^{2}+6h+12)=0^{2}+6(0)+12=12$
