1. The problem is to find the sixth order derivative of the function $y = \arctan(x)$. 2. Recall that the first derivative of $\arctan(x)$ is given by the formula: $$\frac{d}{dx} \arctan(x) = \frac{1}{1+x^2}$$ 3. To find higher order derivatives, we repeatedly differentiate the result. The derivatives of $\arctan(x)$ follow a pattern involving powers of $(1+x^2)$ in the denominator and polynomials in the numerator. 4. The first few derivatives are: - $y' = \frac{1}{1+x^2}$ - $y'' = \frac{-2x}{(1+x^2)^2}$ - $y''' = \frac{6x^2 - 2}{(1+x^2)^3}$ - $y^{(4)} = \frac{-24x^3 + 24x}{(1+x^2)^4}$ - $y^{(5)} = \frac{120x^4 - 240x^2 + 24}{(1+x^2)^5}$ 5. Using this pattern or applying the Leibniz rule and chain rule carefully, the sixth derivative is: $$y^{(6)} = \frac{-720x^5 + 1800x^3 - 720x}{(1+x^2)^6}$$ 6. This expression shows the sixth derivative of $\arctan(x)$ as a rational function with numerator a polynomial of degree 5 and denominator $(1+x^2)^6$. 7. Therefore, the sixth order derivative of $y = \arctan(x)$ is $$\boxed{y^{(6)} = \frac{-720x^5 + 1800x^3 - 720x}{(1+x^2)^6}}$$