Answer
The equation of the tangent line is $y=12x-16$.
Work Step by Step
The slope $m$ of the tangent line of the curve $f(x)$ at $A(a,b)$ is also the slope of the curve at that point, which is calculated by $$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
$$f(t)=t^3\hspace{1cm}A(2,8)$$
1) Find the slope $m$ of the tangent:
$$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
Here $a=2$ and $g(a)=b=8$
$$m=\lim_{h\to0}\frac{f(h+2)-8}{h}=\lim_{h\to0}\frac{(h+2)^3-8}{h}=\lim_{h\to0}\frac{h^3+6h^2+12h+8-8}{h}$$ $$m=\lim_{h\to0}\frac{h^3+6h^2+12h}{h}=\lim_{h\to0}(h^2+6h+12)$$
$$m=0^2+6\times0+12=12$$
2) Find the equation of the tangent line at $A(2,8)$:
The tangent line would have this form: $$y=12x+m$$
Substitute $A(2,8)$ here to find $m$:
$$12\times2+m=8$$ $$24+m=8$$ $$m=-16$$
So the equation of the tangent line is $y=12x-16$.
