Chapter 14: Q. 32 (page 897)

In Problems 21–32, find the derivative of each function at the given number.
$f (x) = \cos (x)$ at $0$

Short Answer

Expert verified

The derivative of the given function at is equal to $0$

Step by step solution

Step 1. Given information

Given function is $f (x) = \cos (x)$

We have to find the derivative of the given function at $0$

Step 2. Definition of the derivative

Let's say a function $y = f (x)$ ,

The derivative of the function at $c$ is defined as

$f' (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$

Step 3. Finding the derivative

By substituting the given function in the derivative definition,

we will get:

$f' (0) = \lim_{x \to 0} \frac{f (x) - f (0)}{x - 0} \Rightarrow f' (0) = \lim_{x \to 0} \frac{\cos (x) - \cos (0)}{x} \Rightarrow f' (0) = \lim_{x \to 0} \frac{\cos (x) - 1}{x}$

Step 4. Simplifying the limit

The expansion of $\cos (x)$ is $\cos (x) = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + . . . . . . . . . .$

Now the limit is:

$f' (0) = \lim_{x \to 0} \frac{(1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + . . . . . . .) - 1}{x} \Rightarrow f' (0) = \underset{x \to 0}{\lim (} - \frac{x}{2!} + \frac{x^{3}}{4!} - \frac{x^{5}}{6!} + . . . . . . .) \Rightarrow f' (0) = 0$

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In Problems 21–32, find the derivative of each function at the given number.
$f (x) = \cos (x)$ at $0$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Definition of the derivative

Step 3. Finding the derivative

Step 4. Simplifying the limit

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In Problems 21–32, find the derivative of each function at the given number.f(x)=cosxat0

Short Answer

Step by step solution

Step 1. Given information

Step 2. Definition of the derivative

Step 3. Finding the derivative

Step 4. Simplifying the limit

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Geometry

Probability and Statistics

Calculus

Applied Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

In Problems 21–32, find the derivative of each function at the given number.
$f (x) = \cos (x)$ at $0$