Define on the domain .

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Define s(x)=sec(x)s(x)=\text{sec}(x) on the domain (−π2,π2)\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right). (a) Express s(x)s(x) as the reciprocal of a trigonometric function and let u(x)u(x) denote that denominator. (b) Derive a general formula for ddx(u(x)−1)\dfrac{d}{dx}\bigl(u(x)^{-1}\bigr) in terms of u(x)u(x) and u′(x)u'(x). Show all steps of your algebra. (c) Apply your result from part (b) to the specific choice of u(x)u(x) identified in part (a) and simplify completely, obtaining s′(x)s'(x) as a product of elementary trigonometric functions. (d) State the largest open interval(s) contained in (−π2,π2)\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right) on which s′(x)s'(x) is continuous, and justify your answer based on your expression from part (c).

Related theory

Browse the problem library