Answer
The equation of the tangent line is $y=2x+3$.
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 $$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
$$y=f(x)=\frac{1}{x^2}\hspace{1cm}A(-1,1)$$
1) Find the slope $m$ of the tangent: $$m=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$$
Here $a=-1$ and $f(a)=b=1$.
$$m=\lim_{h\to0}\frac{\frac{1}{(h-1)^2}-1}{h}=\lim_{h\to0}\frac{\frac{1-(h-1)^2}{(h-1)^2}}{h}$$ $$m=\lim_{h\to0}\frac{1-(h^2-2h+1)}{h(h-1)^2}=\lim_{h\to0}\frac{2h-h^2}{h(h-1)^2}$$ $$m=\lim_{h\to0}\frac{2-h}{(h-1)^2}$$ $$m=\frac{2-0}{(0-1)^2}=\frac{2}{(-1)^2}=\frac{2}{1}=2$$
2) Find the equation of the tangent line at $A(-1,1)$:
The tangent line would have this form: $$y=2x+m$$
Substitute $A(-1,1)$ here to find $m$: $$2\times(-1)+m=1$$ $$-2+m=1$$ $$m=3$$
So the equation of the tangent line is $y=2x+3$.
