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Chapter 4: Problem 12

Find the derivatives of the functions. $$\ln (\cos (x))$$

Short Answer

Expert verified
The derivative of \( \ln(\cos(x)) \) is \( -\tan(x) \).

Step by step solution

01

Recall the Derivative of Natural Logarithm

The derivative of the natural logarithm function \( \ln(u) \) where \( u \) is a function of \( x \) is \( \frac{1}{u} \cdot u' \). In our function, \( u = \cos(x) \).
02

Find the Derivative of \( \cos(x) \)

Compute the derivative of \( \cos(x) \), which is \( -\sin(x) \). So, \( u' = -\sin(x) \).
03

Apply the Chain Rule

Using the chain rule, substitute the function and its derivative into the formula: \( \frac{1}{\cos(x)} \cdot (-\sin(x)) \).
04

Simplify the Expression

Simplify the expression: \( \frac{-\sin(x)}{\cos(x)} \). This reduces to \( -\tan(x) \) because \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
05

Write the Final Derivative

The derivative of \( \ln(\cos(x)) \) is \( -\tan(x) \).

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Key Concepts

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

Natural Logarithm Derivative
The derivative of the natural logarithm function is a fundamental concept in calculus that often appears in simple and complex problems alike. When you take the derivative of a natural logarithm such as \( \ln(u) \), where \( u \) is a function of \( x \), the rule to apply is as follows:
  • The derivative is \( \frac{1}{u} \cdot u' \), where \( u' \) is the derivative of \( u \).
  • In essence, you're dividing 1 by the argument and multiplying by the derivative of the argument itself.
For example, if \( u = \cos(x) \), then using this rule, you would first take the derivative of \( \cos(x) \), which will be part of your final derivative calculation. Remember, the idea is to "unwrap" the natural log by simplifying it using its derivative rule.
Chain Rule
The Chain Rule is a powerful tool in differentiation utilized when finding the derivative of compositions of functions. It is used when you have a function nested inside another function, just like in our example with \( \ln(\cos(x)) \). The basic idea is:
  • If you have a function \( f(g(x)) \), apply the chain rule by differentiating the outer function \( f \) using \( g \) as its argument, then multiply by the derivative of the inner function \( g(x) \).
This can be broken down into these steps:
  • Take the derivative of the outer function while keeping the inner function unchanged.
  • Multiply by the derivative of the inner function.
For our function \( \ln(\cos(x)) \):
  • The outer function is \( \ln(u) \), and its derivative is \( \frac{1}{u} \).
  • The inner function is \( \cos(x) \) with a derivative of \( -\sin(x) \).
  • By applying the chain rule, you multiply \( \frac{1}{\cos(x)} \) by \( -\sin(x) \).
This highlights the beauty of the chain rule in simplifying complex derivatives.
Trigonometric Functions Derivative
When dealing with derivatives, trigonometric functions often play a crucial role in many problems. In calculus, understanding how to differentiate trig functions like \( \cos(x) \) or \( \sin(x) \) is essential:
  • The derivative of \( \cos(x) \) is \( -\sin(x) \). This is a standard result and forms a basis for many calculus operations involving trig functions.
  • Knowing the derivatives of basic trig functions allows us to apply rules like the chain rule effectively when they appear as inner functions in composite functions.
In the exercise at hand, recognizing that \( \cos(x) \)'s derivative is \( -\sin(x) \) simplifies the process substantially. By obtaining this derivative, you can effectively use it in calculations involving the chain rule or even when simplifying complex trigonometric expressions. These basics provide a strong foundation for tackling more involved calculus problems that include trigonometry.

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Most popular questions from this chapter

Find the derivatives of the functions. $$\ln \left(x^{3}+3 x\right)$$

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