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Chapter 3: Problem 25

Find the derivative. \(g(x)=\sin (-x)+\cos (-x)\)

Short Answer

Expert verified
The derivative is \(g'(x) = -\cos(x) - \sin(x)\).

Step by step solution

01

Apply Trigonometric Identities

To find the derivative, first recognize that the sine and cosine of a negative angle can be expressed using the following identities: \(\sin(-x) = -\sin(x)\) and \(\cos(-x) = \cos(x)\). Rewrite the function as \(g(x) = -\sin(x) + \cos(x)\).
02

Differentiate Each Term

Differentiate each term in the rewritten function separately. The derivative of \(-\sin(x)\) with respect to \(x\) is \(-\cos(x)\). The derivative of \(\cos(x)\) with respect to \(x\) is \(-\sin(x)\).
03

Combine the Derivatives

Combine the differentiated terms. The derivative of \(-\sin(x) + \cos(x)\) is \(-\cos(x) - \sin(x)\). Therefore, the derivative \(g'(x) = -\cos(x) - \sin(x)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Identities
Trigonometric identities are crucial shortcuts that help simplify complex trigonometric expressions. They allow you to rewrite functions in a more manageable form. In this exercise, we dealt with negative angles. Use the identities: \(\sin(-x) = -\sin(x)\) and \(\cos(-x) = \cos(x)\). These identities suggest that while the sine function changes its sign with a negative angle, the cosine function does not. They are powerful tools for making differentiation easier. Recognizing these patterns means you can rewrite the function, which sets up the groundwork for differentiation in the next step.
Differentiation
Differentiation is a fundamental concept in calculus. It is the process of finding the rate at which a function is changing at any given point. In this context, the derivative provides insight into the function's behavior, such as its slope and rate of change.

When differentiating a function like \(-\sin(x) + \cos(x)\), handle each term independently. Each differentiation follows structured rules:
  • The derivative of \(-\sin(x)\) is \(-\cos(x)\).
  • The derivative of \(\cos(x)\) is \(-\sin(x)\).


Combining these results gives the complete derivative of the function, which summarizes how each individual part of the original function contributes to its overall rate of change.
Sine and Cosine Functions
Sine and cosine functions are at the heart of trigonometry. They are periodic functions often used to model wave patterns. Understanding their properties is key to both trigonometry and calculus.

These functions have specific derivatives:
  • The derivative of \(\sin(x)\) is \(\cos(x)\).
  • The derivative of \(\cos(x)\) is \(-\sin(x)\).


This means the slope of the sine function at any point \(x\) is given by the cosine of \(x\), and the slope of the cosine function is the negative sine. Recognizing these patterns not only helps in solving derivatives but also in understanding the underlying geometry of the curves.

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