If f(x) = f'(x) = f'(4) = = cos x - 7tanz, then

Calculus: Early Transcendentals
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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...
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### 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.
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.
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