Let for all real .

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

Problem

Let g(x)=sin⁡xg(x)=\sin x for all real xx. (i) Show that sin⁡(x+h)−sin⁡x=2cos⁡ ⁣(x+h2)sin⁡ ⁣(h2)\sin(x+h)-\sin x=2\cos\!\left(x+\dfrac{h}{2}\right)\sin\!\left(\dfrac{h}{2}\right). (ii) Using the result of part (i), evaluate lim⁡h→0sin⁡(x+h)−sin⁡xh\displaystyle\lim_{h\to 0}\frac{\sin(x+h)-\sin x}{h} for an arbitrary real number xx. Give your answer in its simplest trigonometric form. (iii) State the derivative of gg at an arbitrary point xx.

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