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Chapter 3: Problem 20

Find the following derivatives $$\frac{d}{d x}\left(\ln \left(\cos ^{2} x\right)\right)$$

Short Answer

Expert verified
Answer: The derivative of ln(cos^2(x)) with respect to x is -2sin(x)/cos(x).

Step by step solution

01

Identify the outer and inner functions

The given function can be seen as a composition of two functions: the outer function is the natural logarithm (ln), and the inner function is cos^2(x).
02

Apply the chain rule

The chain rule states that if we have a composition of functions, such as f(g(x)), then the derivative of this composition with respect to x can be found as follows: $$\frac{d}{d x}f(g(x)) = f'(g(x)) * g'(x)$$. In our case, the outer function f(x) is ln(x), and the inner function g(x) is cos^2(x).
03

Find the derivatives of the outer and inner functions

First, let's find the derivative of the outer function, f'(x) = d(ln(x))/dx. We know that the derivative of ln(x) is 1/x, so f'(x) = 1/x. Next, let's find the derivative of the inner function, g'(x) = d(cos^2(x))/dx. Since the function cos^2(x) is a composition itself, we need to use the chain rule again. Let h(x) = cos(x), then g(x) = h^2(x). The derivative of h^2(x) is 2h(x) * h'(x). We know that the derivative of cos(x) is -sin(x), so g'(x) = 2cos(x) * (-sin(x)) = -2cos(x)sin(x).
04

Substitute the derivatives into the chain rule and simplify

Now that we have the derivatives for both the outer and inner functions, we can plug them into the chain rule and simplify as follows: $$\frac{d}{d x} ln(cos^2(x)) = \frac{1}{cos^2(x)} * (-2cos(x)sin(x))$$ Multiplying both terms, we get: $$\frac{-2cos(x)sin(x)}{cos^2(x)}.$$ Now, we can simplify by dividing both the numerator and the denominator by cos(x) and obtain: $$-\frac{2sin(x)}{cos(x)}.$$ So, the derivative of ln(cos^2(x)) with respect to x is: $$\frac{d}{d x}\left(\ln \left(\cos ^{2} x\right)\right) = -\frac{2sin(x)}{cos(x)}.$$

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Key Concepts

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

Derivatives
When we talk about derivatives, we鈥檙e essentially discussing how a function changes as its input changes. This is a fundamental concept in calculus, often denoted as \(\frac{d}{dx}\). In simple terms, the derivative gives us the slope of the tangent line to the function at any given point. This slope is the rate at which the function's value is changing.
  • Instantaneous rate of change: The derivative at a point tells us the instantaneous rate of change of the function at that point.
  • Notation: Commonly written as \(f'(x)\) or \(\frac{df}{dx}\).
  • Practical use: Derivatives help in understanding the behavior of functions like motion, growth rates, and optimizing problems.
For example, in the provided exercise, we are finding the derivative of a composed function to see how it changes as \(x\) changes. The derivative process involves not just taking derivatives but recognizing patterns, like products and complicated chains of functions.
Composition of Functions
Composition of functions involves creating a new function by applying one function to the result of another function. In other words, we combine two functions, say \(f(x)\) and \(g(x)\), into a single function \(f(g(x))\). This is akin to placing one function inside another, which happens quite frequently in mathematics and engineering.
  • Structure: The function \(g(x)\) is called the "inner function" and \(f(x)\) is the "outer function".
  • Example: In the exercise, \(\ln(cos^2(x))\) represents a composition where \(\ln(x)\) is the outer function and \(cos^2(x)\) is the inner function.
  • Deriving compositions: The chain rule is used to differentiate such compositions, applying the derivative of both functions step by step.
Understanding composition is key because many real-world scenarios, like nested patterns or layered processing, use this principle.
Natural Logarithm
The natural logarithm, denoted as \(\ln(x)\), is a logarithm with base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. It's a crucial function in mathematics, especially in calculus due to its unique properties and behaviors.
  • Definition: \(\ln(x)\) is the power to which \(e\) must be raised to obtain a number \(x\).
  • Derivative: The derivative of \(\ln(x)\) is \(\frac{1}{x}\), which is crucial when applying the chain rule.
  • Usage: Natural logarithms appear frequently in calculus, solving exponential growth or decay problems, and in any mathematical scenario involving compound interest and fluid dynamics.
In our exercise, \(\ln\) acts as the outer function in a composite derivative problem, demonstrating both its ubiquity and its simplicity in transformation.

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Most popular questions from this chapter

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