If f(x) = 9x^10 tan^−1 x, find f '(x).

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

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