Consider the relation

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Consider the relation sec⁡y=x\sec y = x, where yy is an angle measured in radians and xx satisfies ∣x∣>1|x|>1. This equation defines yy implicitly as a function of xx (that is, y=arcsec⁡xy=\operatorname{arcsec}x). (a) Differentiate both sides with respect to xx to obtain an expression for dydx\dfrac{dy}{dx} in terms of yy and xx. Show each step in your work. (b) Rewrite your result from part (a) so that dydx\dfrac{dy}{dx} is expressed solely in terms of xx and does not involve yy. (c) Justify why an absolute value symbol appears in the simplified derivative and explain its significance in relation to the domain ∣x∣>1|x|>1. (d) Identify the open intervals on the real line where arcsec⁡x\operatorname{arcsec}x is differentiable and explain how your derivative formula supports this conclusion.

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