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Question
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
Solution
\[\text{ We have }: \]
\[f'\left( \frac{\pi}{2} \right) = \lim_{h \to 0} \frac{f\left( \frac{\pi}{2} + h \right) - f\left( \frac{\pi}{2} \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\sin2\left( \frac{\pi}{2} + h \right) - \sin2\left( \frac{\pi}{2} \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\sin(\pi + 2h) - 0}{h}\]
\[ = \lim_{h \to 0} \frac{- \sin2h}{h} \times \frac{2}{2} \]
\[ = \lim_{h \to 0} - \frac{\sin 2h}{2h} \times 2 \]
\[ = - 2\]
\[\]
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