We can use chain rule here. In order to differentiate a function of a function, say y, =f(g(x)), where we have to find (dy)/(dx), we need to do substitute u=g(x), which gives us y=f(u). Then we need to use a formula called Chain Rule, which states that (dy)/(dx)=(dy)/(du)xx(du)/(dx).
Here we have y=tan3x-cot3x. Let u=3x
Then y=tanu-cotu and (dy)/(du)=sec^2u-(-csc^2u)
= sec^2u+csc^2u
Now as u=3x, we have (du)/(dx)=3
and hence (dy)/(dx)=(dy)/(du)xx(du)/(dx)
= (sec^2u+csc^2u)xx3
= (sec^2 3x+csc^2 3x)xx3
= (1/(cos^2 3x)+1/(sin^2 3x))xx3
= (3(cos^2 3x+sin^2 3x))/(cos^2 3xsin^2 3x)
= 3sec^2 3xcsc^2 3x
