The given mathematical problem requires the determination of the first derivative of a function and its evaluation at a specific point.**Problem Statement:**Given the function: $ f(x) = 2 \sin x + 9 \cos x $, find the following:1. The first derivative of the function, denoted as $ f'(x) $.2. The value of the first derivative at $ x = 4 $, denoted as $ f'(4) $.**Solution Steps:**1. **Finding $ f'(x) $**: - The first derivative of the function $ f(x) $ with respect to $ x $ is obtained by differentiating each term of the function. - The derivative of $ 2 \sin x $ is $ 2 \cos x $. - The derivative of $ 9 \cos x $ is $ -9 \sin x $. - Therefore, $ f'(x) = 2 \cos x - 9 \sin x $.2. **Evaluating $ f'(4) $**: - Substitute $ x = 4 $ into the first derivative: - $ f'(4) = 2 \cos 4 - 9 \sin 4 $.In conclusion:- $ f'(x) = \boxed{2 \cos x - 9 \sin x} $- $ f'(4) = \boxed{2 \cos 4 - 9 \sin 4} $This problem illustrates the application of differentiation on trigonometric functions and the evaluation of the derived function at a given point.

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Description: The given mathematical problem requires the determination of the first derivative of a function and its evaluation at a specific point.**Problem Statement:**Given the function: $ f(x) = 2 \sin x + 9 \cos x $, find the following:1. The first deriva

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If f(x) = 2 sin æ + 9 cos a, then f'(x) = f'(4) =|

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.5: Transformation Of Functions

Problem 5SE: How can you determine whether a function is odd or even from the formula of the function?

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The given mathematical problem requires the determination of the first derivative of a function and its evaluation at a specific point.

**Problem Statement:**

Given the function: $ f(x) = 2 \sin x + 9 \cos x $, find the following:

1. The first derivative of the function, denoted as $ f'(x) $.
2. The value of the first derivative at $ x = 4 $, denoted as $ f'(4) $.

**Solution Steps:**

1. **Finding $ f'(x) $**:
- The first derivative of the function $ f(x) $ with respect to $ x $ is obtained by differentiating each term of the function.
- The derivative of $ 2 \sin x $ is $ 2 \cos x $.
- The derivative of $ 9 \cos x $ is $ -9 \sin x $.
- Therefore, $ f'(x) = 2 \cos x - 9 \sin x $.

2. **Evaluating $ f'(4) $**:
- Substitute $ x = 4 $ into the first derivative:
- $ f'(4) = 2 \cos 4 - 9 \sin 4 $.

In conclusion:

- $ f'(x) = \boxed{2 \cos x - 9 \sin x} $
- $ f'(4) = \boxed{2 \cos 4 - 9 \sin 4} $

This problem illustrates the application of differentiation on trigonometric functions and the evaluation of the derived function at a given point.

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