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Question
Find the derivative of f (x) = cos x at x = 0
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Solution
We have:
\[f'(x) = \lim_{h \to 0} \frac{f(0 + h) - f(0)}{h}\]
\[ = \lim_{h \to 0} \frac{f(h) - f(0)}{h}\]
\[ = \lim_{h \to 0} \frac{\cosh - \cos0}{h}\]
\[ = \lim_{h \to 0} \frac{\cosh - 1}{h}\]
\[ {= \lim}_{h \to 0} \frac{- 2 \sin^2 \frac{h}{2}}{h}\]
\[ {= \lim_{h \to 0} \frac{- 2 \sin^2 \frac{h}{2}}{\frac{h^2}{4}}}_{} \times \frac{h}{4}\]
\[ = {= \lim_{h \to 0} - 1}_{} \times \frac{h}{2}\]
\[ = 0\]
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