Answer
The equation for the tangent line for $x=2$ is $$y=\frac{135}{16}x-\frac{27}{2}$$
Work Step by Step
The equation for the tangent line at $x_0$ is:
$$y-f(x_0)=f'(x_0)(x-x_0)$$
Here is $x_0=2$ and $f(x)=(x-\frac{1}{x})^3$.
$$f'(x)=\Big(\Big(x-\frac{1}{x}\Big)^3\Big)'=3\Big(x-\frac{1}{x}\Big)^2\Big(x-\frac{1}{x}\Big)'=3\Big(x-\frac{1}{x}\Big)^2(1-(-\frac{1}{x^2}))=3\Big(x-\frac{1}{x}\Big)^2\frac{x^2+1}{x^2}$$
Now for $x=2$ we have:
$$f(2)=(2-\frac{1}{2})^3=\frac{27}{8}$$
$$f'(2)=3\Big(2-\frac{1}{2}\Big)^2\frac{2^2+1}{2^2}=3\frac{9}{4}\cdot\frac{5}{4}=\frac{135}{16}$$
So the equation for the tangent line at $x=2$ is:
$$y-\frac{27}{8}=\frac{135}{16}(x-2)\Rightarrow y=\frac{135}{16}x-\frac{135}{8}+\frac{27}{8}\Rightarrow y=\frac{135}{16}x-\frac{27}{2}$$
