Home » Blog » Establish the formula for the derivative of arctan x

Establish the formula for the derivative of arctan x

by RoRi

Establish the following formula for the derivative of \arctan x is correct,

    \[ D(\arctan x) = \frac{1}{1+x^2}, \qquad \text{for all } x \in \mathbb{R}. \]

For x \in \mathbb{R} let

    \[ y = \arctan x \quad \implies \quad x = \tan y. \]

Then we know

    \[ \frac{dx}{dy} = \sec^2 y. \]

Therefore, by Theorem 6.7 (p. 252 of Apostol) we have,

    \begin{align*} \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} && \implies && \sec^2 y &= \frac{1}{\frac{dy}{dx}} \\ && \implies && \frac{dy}{dx} &= \frac{1}{\sec^2 y} \\ && \implies && D(\arctan x) &= \frac{1}{\sec^2 y}. \end{align*}

Using a trig identity for tangent and secant we have

    \[ \sec^2 y = 1 + \tan^2 y = 1 + x^2 \]

since x = \tan y. Therefore we conclude,

    \[ D(\arctan x) = \frac{1}{1+x^2}.\]

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):