Compute the derivative of sin x. | vedclass

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(A) Let $f(x) = \sin x$. Then by the definition of the derivative:$\frac{df(x)}{dx} = \lim_{h 	o 0} \frac{f(x+h) - f(x)}{h} = \lim_{h 	o 0} \frac{\s

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$\cos x$

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(A) Let $f(x) = \sin x$. Then by the definition of the derivative:$\frac{df(x)}{dx} = \lim_{h 	o 0} \frac{f(x+h) - f(x)}{h} = \lim_{h 	o 0} \frac{\sin(x+h) - \sin(x)}{h}$Using the trigonometric identity $\sin A - \sin B = 2 \cos\left(\frac{A+B}{2}ight) \sin\left(\frac{A-B}{2}ight)$:$= \lim_{h 	o 0} \frac{2 \cos\left(\frac{2x+h}{2}ight) \sin\left(\frac{h}{2}ight)}{h}$$= \lim_{h 	o 0} \cos\left(x + \frac{h}{2}ight) \cdot \lim_{h 	o 0} \frac{\sin(h/2)}{h/2}$Since $\lim_{h 	o 0} \cos(x + h/2) = \cos x$ and $\lim_{	heta 	o 0} \frac{\sin 	heta}{	heta} = 1$,we get:$= \cos x \cdot 1 = \cos x$

Compute the derivative of $\sin x$.

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