Question

For the following exercises, find the derivative \(d y / d x\). (You can use a calculator to plot the function and the derivative to confirm that it is correct.) $$ \text { [T] } y=\ln \left(\cos ^{2} x\right) $$

Short answer

The derivative is \( \frac{dy}{dx} = -2 \tan x \).

Step-by-step solution

Identify the function structure

The given function is \( y = \ln(\cos^2 x) \). Notice that this is a composition of functions, specifically \( \ln(u) \) where \( u = \cos^2 x \). Thus, we need to apply the chain rule to differentiate it.

Apply the Chain Rule

According to the chain rule, the derivative of \( y = \ln(u) \) with respect to \( x \) is \( \frac{d}{dx}[\ln(u)] = \frac{1}{u} \cdot \frac{du}{dx} \). Here, \( u = \cos^2 x \), so we first need to find \( \frac{du}{dx} \).

Differentiate \( u = \cos^2 x \)

\( u = \cos^2 x \) can be rewritten as \( u = (\cos x)^2 \). To differentiate \( u \), use the chain rule for powers: \( \frac{d}{dx}[u] = 2 \cos x \cdot \frac{d}{dx}[\cos x] = 2 \cos x \cdot (-\sin x) = -2 \cos x \sin x \).

Differentiate \( y = \ln(u) \)

Substitute \( u = \cos^2 x \) and \( \frac{du}{dx} = -2 \cos x \sin x \) into the derivative formula: \( \frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} = \frac{1}{\cos^2 x} \cdot (-2 \cos x \sin x) \).

Simplify the derivative

Simplify the expression: \( \frac{dy}{dx} = -2 \sin x \cdot \frac{\cos x}{\cos^2 x} = -2 \sin x / \cos x = -2 \tan x \), where \( \tan x = \frac{\sin x}{\cos x} \). Thus, \( \frac{dy}{dx} = -2 \tan x \).

Key concepts

Chain Rule

The chain rule is a fundamental tool in calculus used for finding the derivative of a composition of functions. When you have a function inside another function, like \(y = \ln(\cos^2 x)\), you need to use the chain rule to differentiate it correctly. The basic idea is to differentiate the outer function while considering the inner function as a single entity. Then, multiply this result by the derivative of the inner function.

For our problem, recognizing that \(u = \cos^2 x\) is crucial. Differentiate the outer function, which in our case is \ln(u)\ or \frac{1}{u}\, and then multiply it by the derivative of \(u\), which is the inner function. This step-by-step approach ensures the correct application of differentiation rules to complex functions.

Derivative

In calculus, the derivative of a function measures how the function's value changes as its input changes. It's like a mathematical way of finding the slope or rate of change at any particular point.

When you want to find the derivative \frac{dy}{dx}\, you are essentially determining how much \(y\) changes with a tiny change in \(x\). Doing so for complicated functions often requires rules like the chain rule to decompose the function into manageable parts, differentiating each one separately before combining them appropriately. For \(y = \ln(\cos^2 x)\), the derivative tells us how the natural logarithm of the square of cosine reacts to changes in \(x\).

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are crucial in calculus. These functions often appear in problems involving angles, periodic phenomena, or whenever you're dealing with circular motion.

The cosine function, used in our case, has a derivative of \-\sin(x)\. The transformation of the cosine function, such as squaring it \(\cos^2(x)\), involves additional steps in finding derivatives because it becomes a compound function. Recognizing \(\cos^2 x\) as \(\left(\cos x\right)^2\) helps us apply the power rule and combine it with the chain rule for accurate differentiation.

Understanding these foundational trigonometric behaviors simplifies dealing with these types of calculus problems.

Logarithmic Differentiation

Logarithmic differentiation is a technique used particularly when dealing with functions of the form \(y = \ln(f(x))\), but can be expanded to other forms like products or powers of functions. It's incredibly useful when the function involves logarithms naturally or can benefit from the properties of logarithms to make differentiation easier.

For our exercise, differentiating \(y = \ln(\cos^2 x)\) inherently requires applying properties of logarithms. The derivative of a logarithmic function \(\ln(u)\) is \(\frac{1}{u} \, \frac{du}{dx}\). Using this property simplifies the differentiation process, reducing complex problems into simpler fractions or expressions. Always remember to apply the derivative of the inside function, ensuring that each piece of the puzzle is accounted for.