Consider the function

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

Problem

Consider the function g(x)=tan⁡(5x)g(x)=\tan(5x), which models the angular displacement (in radians) of a rapidly oscillating device as a function of time xx (in seconds). (a) Derive an explicit expression for g′(x)g'(x). (b) Determine g′′(x)g''(x) and simplify completely. (c) Evaluate g′′′ ⁣(π4)g'''\!\left(\dfrac{\pi}{4}\right) and express your answer as an exact real number.

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