Home » Blog » Find the derivative of x1/2 – arctan (x1/2)

Find the derivative of x1/2 – arctan (x1/2)

by RoRi

Find the derivative of the function

    \[ f(x) = \sqrt{x} - \arctan \sqrt{x}. \]

Using the formula for the derivative of arctangent and the chain rule we have

    \begin{align*} f'(x) &= \frac{1}{2\sqrt{x}} - \left( \frac{1}{2\sqrt{x}} \right) \left( \frac{1}{1+(\sqrt{x})^2} \right) \\ &= \left( \frac{1}{2 \sqrt{x}} \right) \left( 1- \frac{1}{1+x} \right) \\ &= \left( \frac{1}{ 2\sqrt{x}} \right) \left( \frac{x}{1+x} \right) \\ &= \frac{\sqrt{x}}{2(1+x)}. \end{align*}

This is valid for x \geq 0.

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