MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Let p(x)=tan(3x)+sec(2x) for all real x in the domains of the functions.
Answer each of the following, expressing each derivative in simplest exact form.
(1) Derive a closed‐form expression for p′(x). Clearly show all steps in your differentiation.
(2) Determine p′′(x) and simplify your result so that no trigonometric function appears with an argument that can be factored by a common constant.
(3) Obtain p′′′(x), writing your final answer as a sum of products of powers of sec(2x) and tan(2x) together with powers of sec(3x) and tan(3x).
(4) Evaluate p′′′(0) exactly and interpret the sign of this value in terms of the local concavity behavior of p′′ near x=0. State your reasoning clearly.