f(x) = 9x^10 tan^−1 x is a product: f(x) = (9) * (x^10) * (arctan x).
Use the product rule: (d/dx) (u*v) = u*dv/dx + v*du/dx and the inverse tangent rule:
(d/dx)(arctan x) = 1 / (1 + x^2).
Then the desired derivative is:
1
f '(x) = 9 [x^10*----------- + arctan x*10x^9 ]
1+x^2
Note that x^9 can be factored out:
x
f '(x) = 9*x^9 [ ----------- + 10arctan x ]
1+x^2
