Chapter 14: Q. 31 (page 897)

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

Short Answer

Expert verified

The derivative of the given function at $0$ is equal to $1$ .

Step by step solution

Step 1. Given information

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

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:

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

Step 4. Simplifying the limit

The expansion of $\sin (x)$ is:

$\sin (x) = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + . . . . . . . .$

Now the limit is:

localid="1647086780847">f'(0)=limx→0x-x33!+x55!-x77!+.......x⇒f'(0)=limx→01-x23!+x45!-x67!+.......⇒f'(0)=1

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Discrete Mathematics

Pure Maths

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

In Problems 21–32, find the derivative of each function at the given number.f(x)=sinxat0

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

Decision Maths

Probability and Statistics

Discrete Mathematics

Pure Maths

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

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