Problem 17 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Differentiate the given function. $$f(x)=\ln \left(\cos ^{2} x\right)$$

Step-by-Step Solution

Answer

The derivative of \( f(x) = \ln ( \cos^2(x) ) \) is \( f'(x) = -2 \tan(x) \).

Step 1: Recognize the Outer Function

The given function is composed of two nested functions. The outer function is the natural logarithm function, which is \(f(u) = \ln u\).

Step 2: Identify the Inner Function

The inner function inside the logarithm is \(u = \cos^2(x)\).

Step 3: Differentiate the Outer Function

Differentiate \(f(u) = \ln u\) with respect to \(u\). We get \( \frac{d}{du} [\ln u] = \frac{1}{u}\).

Step 4: Differentiate the Inner Function

Now, differentiate the inner function \(u = \cos^2(x)\). Since \(u = (\cos(x))^2\), use the chain rule: \( \frac{d}{dx} [ \cos^2(x) ] = 2 \cos(x) \frac{d}{dx} [ \cos(x) ] = 2 \cos(x) \cdot ( -\sin(x) )= -2 \cos(x) \sin(x) = - \sin(2x)\).

Step 5: Combine Results Using the Chain Rule

Apply the chain rule: \( \frac{d}{dx} [ \ln( \cos^2(x) ) ] = \frac{d}{du} [ \ln( u ) ] \cdot \frac{d}{dx} [ \cos^2(x) ] = \frac{1}{ \cos^2(x) } \cdot ( - \sin(2x) ) = - \frac{ \sin(2x)}{ \cos^2(x) }\).

Step 6: Simplify the Expression

Simplify the expression further. \( - \frac{ \sin(2x)}{ \cos^2(x) } = - \frac{ 2 \sin(x) \cos(x) }{ \cos^2(x) } = -2 \frac{ \sin(x) }{ \cos(x) } = -2 \tan(x) \).

Key Concepts

Chain Rule

To differentiate a function of another function, we use the chain rule. In simpler terms, it helps us handle 'nested' functions—functions inside other functions.
In our exercise, we have \(f(x) = \ln(\cos^2(x))\). Here, \(\ln(\cos^2(x))\) has an outer function, \(\ln(u)\)\, and an inner function, \(u = \cos^2(x)\).
The chain rule says to first differentiate the outer function, then multiply it by the derivative of the inner function.

• First, differentiate \(\ln(u) \) which gives us \( \frac{1}{u} \).
• Then, differentiate the inner function \ (u = \cos^2(x)) ∶, gotten by \(2 \cos(x) \cdot -\sin(x)\) which simplifies to \(-2 \cos(x) \sin(x) = -sin(2x)\).

Combining these results, we use chain rule: \ \frac{d}{dx} \[\ln(\cos^2(x))\] = \frac{1}{\cos^2(x)} \cdot -sin(2x)\.
Powerful, ensures we don't miss any details while differentiating, making sense of complex functions easily.

Logarithmic Differentiation

Logarithmic differentiation is especially helpful when dealing with functions like our example, \(f(x) = \ln (\cos^2(x))\).
This technique simplifies differentiation by leveraging properties of logarithms.
The main idea: take the natural logarithm of the given function and then differentiate.

• When differentiating, use \( \frac{d}{dx} \ln u = \frac{1}{u} \cdot \frac{du}{dx} \).
• In our exercise, \(\ln (\cos^2(x))\) already has a natural logarithm.

Once identified, proceed with the chain rule, ensuring you account for both parts of the composite function.
This method significantly reduces complexity, especially when managing products, powers, and quotients inside functions.

Trigonometric Functions

To differentiate our function, understanding trigonometric functions and their derivatives is crucial.
Here, we deal specifically with cosine and sine:

• \(\cos(x)\) and its derivative, \( -\sin(x)\).
• Recognize expressions like \(\cos^2(x)\) and use trigonometric identities like \ \cos^2(x) + \sin^2(x) = 1 \.

In our solution, while differentiating \(\cos^2(x)\), we did:
\( \frac{d}{dx}(\cos^2(x)) = 2 \cos(x) \cdot ( -\sin(x) ) = - \sin(2x) \).
Knowing these derivatives and identities equips you to better tackle similar problems, transforming complexity into simplicity with practiced familiarity.