truein truein Math 16A

Kouba

The Derivative of Sin x

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FACTS: 1.) $${ \lim_{w \to 0} { \sin w \over w } = 1 }$$ truein 2.) $${ \lim_{w \to 0} { \cos w - 1 \over w } = 0 }$$ truein 3.) $\sin(A+B) = \sin A \cos B + \cos A \sin B$

truein truein truein Let $f(x) = \sin x$ . Then $f'(x) = \displaystyle{ \lim_{ h \to 0} { f(x+ h) - f(x) \over h } }$

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$\ \ \ \ \ = \displaystyle{ \lim_{ h \to 0} { \sin(x+ h) - \sin x \over h } }$

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$\ \ \ \ \ = \displaystyle{ \lim_{ h \to 0} { \sin x \cdot \cos h + \cos x \cdot \sin h - \sin x \over h } }$

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$\ \ \ \ \ = \displaystyle{ \lim_{ h \to 0} { \sin x \cdot ( \cos h - 1)+ \cos x \cdot \sin h \over h } }$

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$\ \ \ \ \ = \displaystyle{ \lim_{ h \to 0} \Bigg\{ \sin x \cdot \Big( { \cos h -1 \over h}\Big) + \cos x \cdot \Big( { \sin h \over h} \Big) \Bigg\} }$

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$\ \ \ \ \ = \sin x \cdot (0) + \cos x \cdot (1)$

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$\ \ \ \ \ = \cos x$