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Magazine Chapter 3: Problem 22
Find \(d y / d x\) $$ y=\ln (\cos x) $$
Short Answer
Expert verified
The derivative is \( \frac{dy}{dx} = -\tan x \).
Step by step solution
01
Understand the Function
We are tasked with finding the derivative of the function \( y = \ln(\cos x) \). This involves using our knowledge of logarithmic and trigonometric differentiation rules.
02
Identify the Chain Rule Structure
The function \( y = \ln(\cos x) \) is a composition of two functions: the outer function \( u = \ln u \) and the inner function \( u = \cos x \). We need to apply the chain rule to find the derivative.
03
Differentiate the Outer Function
According to the chain rule, first differentiate the outer function. The derivative of \( \ln u \) with respect to \( u \) is \( \frac{1}{u} \). Therefore, \( \frac{d}{du}(\ln u) = \frac{1}{u} \), where \( u = \cos x \).
04
Differentiate the Inner Function
The derivative of the inner function \( u = \cos x \) with respect to \( x \) is \( \frac{d}{dx}(\cos x) = -\sin x \).
05
Apply the Chain Rule
Combine the derivatives from steps 3 and 4 using the chain rule: \( \frac{dy}{dx} = \frac{d}{du}(\ln u) \cdot \frac{du}{dx} \). Substitute back the expressions to get: \( \frac{dy}{dx} = \frac{1}{\cos x} \cdot (-\sin x) \).
06
Simplify the Expression
This expression can be simplified as: \( \frac{dy}{dx} = -\frac{\sin x}{\cos x} = -\tan x \), because \( \frac{\sin x}{\cos x} \) is the same as \( \tan x \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chain Rule
The chain rule is a fundamental concept in calculus for finding derivatives of composite functions. When you have a function nested inside another, the chain rule helps you differentiate them smoothly. In our exercise, the function is given as \( y = \ln(\cos x) \), which is a composite of two functions: the natural logarithm \( \ln(u) \) and the trigonometric function \( u = \cos x \).
To apply the chain rule, follow these steps:
To apply the chain rule, follow these steps:
- Differentiating the outer function: Start with \( \ln(u) \). Its derivative is \( \frac{1}{u} \).
- Differentiating the inner function: This is \( \cos x \), whose derivative is \(-\sin x\).
- Combine these using the chain rule: Multiply the derivative of the outer function by the derivative of the inner function. This results in \( \frac{1}{\cos x} \,\, (-\sin x) \).
Trigonometric Differentiation
Understanding trigonometric differentiation is key when dealing with functions involving sine, cosine, and other similar functions. In our given function, we have \( \cos x \), which is a common trigonometric function.
When differentiating \( \cos x \), the result is \( -\sin x \). Here's why:
When differentiating \( \cos x \), the result is \( -\sin x \). Here's why:
- The cosine function's slope at any point gives us its angle's rate of change.
- The negative sign appears because, as the cosine value decreases in a clockwise rotation on the unit circle, the sine value increases negatively.
Natural Logarithm
The natural logarithm, often denoted as \( \ln \), is a logarithm to the base \( e \), where \( e \approx 2.718 \). It's a valuable tool in calculus for simplifying the differentiation of logarithmic functions.
In the exercise, the function is \( \ln(\cos x) \), demonstrating the composition of a natural logarithm with a trigonometric function. When differentiating the natural logarithm of a function, use these rules:
In the exercise, the function is \( \ln(\cos x) \), demonstrating the composition of a natural logarithm with a trigonometric function. When differentiating the natural logarithm of a function, use these rules:
- The derivative of \( \ln(u) \) with respect to its argument \( u \) is \( \frac{1}{u} \).
- You must multiply it by the derivative of \( u \) itself (e.g., if \( u = \cos x \), multiply by \(-\sin x\)).
