Let $(a, b)$ be an arbitrary point on the graph of $y=\frac1x$ ($x>0$). Prove that the area of the triangle formed by the tangent line through $(a,b)$ and the coordinate axes is $2$ square units.
I know that I need to use derivatives and I've already come up with $f'(a)=\frac{1}{2a^2}$, but I am not sure how to use that or where to go from there.
Thanks!
we have $f'(x)=-1/x^2$ thus we get the slope of our tangent as $m=-1/a^2$ and the tangent line is given by $y=-1/a^2x+n$ For the point at the curve we have $x=a,y=1/a$ and our tangentline is given by $y=-1/a^2x+2/a$. therefore the area is $\frac{2}{a}2a\frac{1}{2}=2$