Chapter 2: Q. 83 (page 235)

Use the definition of the derivative, a trigonometric identity, and known trigonometric limits to prove that $\frac{d}{d x} (\cos x) = - \sin x$

Short Answer

Expert verified

The trigonometric limit $\frac{d}{d x} (\cos x) = - \sin x$ has been proved.

Step by step solution

Step 1. Given Information.

The trigonometric identity:

$\frac{d}{d x} (\cos x) = - \sin x$

Step 2. Definition of derivative.

From the definition derivative,

$f' (x) = \underset{h \to 0}{l i m} \frac{f (x + h) - f (x)}{h}$

Step 3. Obtain the derivative for the given function.

From the definition of derivative,

$f' (\cos x) = \underset{h \to 0}{l i m} [\frac{\cos (x + h) - \cos x}{h}] = \underset{h \to 0}{l i m} [\frac{(\cos x \cos h - \sin x . \sin h) - \cos x}{h}] = \underset{h \to 0}{l i m} [\frac{\cos x (\cos h - 1) - \sin x \sin h}{h}] = \underset{h \to 0}{l i m} [\cos x \frac{\cos h - 1}{h} - \sin x \frac{\sin h}{h}]$

Step 4. Apply the limits.

So the function become,

$f' (\cos x) = \cos x \underset{h \to 0}{l i m} \frac{\cos h - 1}{h} - \sin x \underset{h \to 0}{l i m} \frac{\sin h}{h} = \cos x (0) - \sin x (1) = - \sin x$

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Use the definition of the derivative, a trigonometric identity, and known trigonometric limits to prove that $\frac{d}{d x} (\cos x) = - \sin x$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Definition of derivative.

Step 3. Obtain the derivative for the given function.

Step 4. Apply the limits.

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Use the definition of the derivative, a trigonometric identity, and known trigonometric limits to prove that ddx(cosx)=-sinx

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Definition of derivative.

Step 3. Obtain the derivative for the given function.

Step 4. Apply the limits.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Use the definition of the derivative, a trigonometric identity, and known trigonometric limits to prove that $\frac{d}{d x} (\cos x) = - \sin x$