Active Calculus 2nd Ed

Let \(r(x) = \arctan(x)\text{.}\) Use the relationship between the arctangent and tangent functions to rewrite this equation using only the tangent function.

Differentiate both sides of the equation you found in (a). Solve the resulting equation for \(r'(x)\text{,}\) writing \(r'(x)\) as simply as possible in terms of a trigonometric function evaluated at \(r(x)\text{.}\)

Recall that \(r(x) = \arctan(x)\text{.}\) Update your expression for \(r'(x)\) so that it only involves trigonometric functions and the independent variable \(x\text{.}\)

Introduce a right triangle with angle \(\theta\) so that \(\theta = \arctan(x)\text{.}\) What are the three sides of the triangle?

In terms of only \(x\) and \(1\text{,}\) what is the value of \(\cos(\arctan(x))\text{?}\)

Use the results of your work above to find an expression involving only \(1\) and \(x\) for \(r'(x)\text{.}\)