Let for all where the expression is defined.

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

Problem

Let q(x)=(cot⁡x)2q(x)=\bigl(\cot x\bigr)^{2} for all xx where the expression is defined. (1) Find ddx(cot⁡x)\dfrac{d}{dx}(\cot x) and express your answer using a single trigonometric function. (2) Using your result from part (a) and appropriate differentiation techniques, compute q′(x)q'(x). (3) Show algebraically that q′(x)=−2cot⁡xcsc⁡2xq'(x)=-2\cot x\csc^{2}x and explain each simplification step clearly.

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