Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
![### Calculating the Derivative
Given the function \( f(x) = \cos x - 7 \tan x \), we are tasked with finding the derivative \( f'(x) \) and evaluating it at \( x = 4 \).
#### Steps:
1. **Find \( f'(x) \):**
- The derivative of \( \cos x \) with respect to \( x \) is \( -\sin x \).
- The derivative of \( \tan x \) with respect to \( x \) is \( \sec^2 x \).
- Therefore, the derivative of \( 7 \tan x \) is \( 7 \sec^2 x \).
Hence, combining these results:
\[
f'(x) = -\sin x - 7 \sec^2 x
\]
2. **Evaluate \( f'(4) \):**
- Substitute \( x = 4 \) into the derivative:
\[
f'(4) = -\sin(4) - 7 \sec^2(4)
\]
- Calculate the values of \( \sin(4) \) and \( \sec^2(4) \) to find \( f'(4) \) (use a calculator or known trigonometric tables).
This problem requires understanding of basic differentiation rules and trigonometric derivatives to complete.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F16b88384-b27c-4aea-b4d5-cecb5406683e%2Fb99155c8-e45a-4d2b-84c2-0686ebabc572%2F2g9gaas_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Derivative
Given the function \( f(x) = \cos x - 7 \tan x \), we are tasked with finding the derivative \( f'(x) \) and evaluating it at \( x = 4 \).
#### Steps:
1. **Find \( f'(x) \):**
- The derivative of \( \cos x \) with respect to \( x \) is \( -\sin x \).
- The derivative of \( \tan x \) with respect to \( x \) is \( \sec^2 x \).
- Therefore, the derivative of \( 7 \tan x \) is \( 7 \sec^2 x \).
Hence, combining these results:
\[
f'(x) = -\sin x - 7 \sec^2 x
\]
2. **Evaluate \( f'(4) \):**
- Substitute \( x = 4 \) into the derivative:
\[
f'(4) = -\sin(4) - 7 \sec^2(4)
\]
- Calculate the values of \( \sin(4) \) and \( \sec^2(4) \) to find \( f'(4) \) (use a calculator or known trigonometric tables).
This problem requires understanding of basic differentiation rules and trigonometric derivatives to complete.
Expert Solution

Step 1
To differentiate the given function
Step by step
Solved in 2 steps with 1 images
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