Chapter 2: Q. 3 (page 232)

What limit facts and trigonometric identities are used in
the proof that $\frac{d}{d x} (\sin x) = \cos x ?$

Short Answer

Expert verified

limit facts $\lim_{h \to 0} \frac{(\cos h - 1)}{h} = 0 & \lim_{h \to 0} \frac{\sin h}{h} = 1$ and sum identity for the sine $\sin (a + b) = \sin a \cos b + \cos a \sin b$ are used in the proof of $\frac{d}{d x} (\sin x) = \cos x .$

Step by step solution

Step 1. Given information.

the given formula is $\frac{d}{d x} (\sin x) = \cos x .$

Step 2. proof.

Use the definition of the derivative to derivate $f (x) = \sin x .$

$\frac{d}{d x} \sin x = \lim_{h \to 0} \frac{\sin (x + h) - \sin x}{h}$

Apply sum identity for the sine , $\sin (a + b) = \sin a \cos b + \cos a \sin b .$

role="math" localid="1648766873735" $\frac{d}{d x} \sin x = \lim_{h \to 0} \frac{\sin x \cos h + \cos x \sin h - \sin x}{h} \frac{d}{d x} \sin x = \lim_{h \to 0} \frac{\sin x (\cos h - 1) + \cos x \sin h}{h} \frac{d}{d x} \sin x = \sin x (\lim_{h \to 0} \frac{(\cos h - 1)}{h}) + \cos x (\lim_{h \to 0} \frac{\sin h}{h})$

Use limit fact $\lim_{h \to 0} \frac{(\cos h - 1)}{h} = 0 & \lim_{h \to 0} \frac{\sin h}{h} = 1$

$\frac{d}{d x} \sin x = \sin x (0) + \cos x (1) \frac{d}{d x} \sin x = \cos x$

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What limit facts and trigonometric identities are used in
the proof that $\frac{d}{d x} (\sin x) = \cos x ?$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. proof.

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Short Answer

Step by step solution

Step 1. Given information.

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Most popular questions from this chapter

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What limit facts and trigonometric identities are used in
the proof that $\frac{d}{d x} (\sin x) = \cos x ?$