Answer
The equation of the tangent line is $y=-2x+6$
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 $$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
$$g(x)=\frac{8}{x^2}\hspace{1cm}A(2,2)$$
1) Find the slope $m$ of the tangent:
$$m=\lim_{h\to0}\frac{g(a+h)-g(a)}{h}$$
Here $a=2$ and $g(a)=b=2$
$$m=\lim_{h\to0}\frac{g(h+2)-2}{h}=\lim_{h\to0}\frac{\frac{8}{(h+2)^2}-2}{h}=\lim_{h\to0}\frac{\frac{8-2(h+2)^2}{(h+2)^2}}{h}$$ $$m=\lim_{h\to0}\frac{8-2(h^2+4h+4)}{h(h+2)^2}=\lim_{h\to0}\frac{8-2h^2-8h-8}{h(h+2)^2}$$ $$m=\lim_{h\to0}\frac{-2h^2-8h}{h(h+2)^2}=\lim_{h\to0}\frac{-2h-8}{(h+2)^2}$$
$$m=\frac{-2\times0-8}{(0+2)^2}=\frac{-8}{4}=-2$$
2) Find the equation of the tangent line at $A(2,2)$:
The tangent line would have this form: $$y=-2x+m$$
Substitute $A(2,2)$ here to find $m$:
$$-2\times2+m=2$$ $$-4+m=2$$ $$m=6$$
So the equation of the tangent line is $y=-2x+6$.
