Prove that the derivative of the function at any point equals

MATHEMATICS · Subjective · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Prove that the derivative of the function f(x)=sin⁡xf(x)=\sin x at any point x=ax=a equals cos⁡a\cos a. To do so, proceed as follows: (i) State the two standard trigonometric limits required for the proof. (ii) Starting from the definition f′(a)=lim⁡h→0sin⁡(a+h)−sin⁡ahf'(a)=\displaystyle \lim_{h\to 0}\dfrac{\sin(a+h)-\sin a}{h}, simplify the numerator using a suitable trigonometric identity. (iii) Evaluate the resulting limit to establish that f′(a)=cos⁡af'(a)=\cos a.

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