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Question
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
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Solution
\[\left( i \right) \text{ We have }: \]
\[f'\left( \frac{\pi}{2} \right) = \lim_{h \to 0} \frac{f\left( \frac{\pi}{2} + h \right) - f\left( \frac{\pi}{2} \right)}{h}\]
\[ = \lim_{h \to 0} \frac{sin\left( \frac{\pi}{2} + h \right) - sin\left( \frac{\pi}{2} \right)}{h}\]
\[ = \lim_{h \to 0} \frac{cos h - 1}{h}\]
\[ {= \lim}_{h \to 0} \frac{- 2 \sin^2 \frac{h}{2}}{h}\]
\[ {= \lim_{h \to 0} \frac{- 2 \sin^2 \frac{h}{2}}{\frac{h^2}{4}}}_{} \times \frac{h}{4}\]
\[ {= \lim_{h \to 0} - 1}_{} \times \frac{h}{2}\]
\[ = 0\]
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