The following prompts in this activity will lead you to develop the derivative of the inverse tangent function.

(a) Let r(x)=arctan(x). Use the relationship between the arc tangent and tangent functions to rewrite this equation using only the tangent function.

(b)Differentiate both sides of the equation you found in (a). Solve the result in gequation fo rr′(x), writing r′(x) as simply as possible in terms of a trigonometric function evaluated at r(x).

(c) Recall that r(x)=arctan(x). Update your expression for r′() so that it only involves trigonometric functions and the independent variable x.

(d) Introduce a right triangle with angle θ so that θ=arctan(x). What are the three sides of the triangle?

(e) In terms of only x and 1, what is the value of cos(arctan(x))?