Question

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

Short answer

Answer: The derivative of the function $$\ln(\cos^2 x)$$ with respect to x is $$-2 \sin x$$.

Step-by-step solution

Identify the inner and outer functions

The given function is $$\ln(\cos^2 x)$$. Here, the outer function is the natural logarithm function, and the inner function is the cosine squared function: $$\cos^2 x$$.

Apply the chain rule

The chain rule states that the derivative of the composite function is the derivative of the outer function times the derivative of the inner function. In our case, the chain rule is as follows: $$\frac{d}{dx}\left(\ln(\cos^2 x)\right) = \frac{d(\ln(\cos^2 x))}{d(\cos^2 x)} \cdot \frac{d(\cos^2 x)}{dx}$$

Compute the derivative of the outer function

Let's find the derivative of the outer function with respect to the inner function: $$\frac{d(\ln(\cos^2 x))}{d(\cos^2 x)} = \frac{1}{\cos^2 x}$$

Compute the derivative of the inner function

Next, we need to find the derivative of the inner function with respect to x: $$\frac{d(\cos^2 x)}{dx} = \frac{d}{dx}(\cos x \cdot \cos x) = 2 \cos x \cdot (-\sin x) = -2 \cos x \sin x$$

Multiply and simplify

Now, we can multiply the derivatives we found in Step 3 and Step 4 and simplify the expression: $$\frac{d}{dx}\left(\ln(\cos^2 x)\right) = \frac{1}{\cos^2 x} \cdot (-2 \cos x \sin x) = -2 \sin x$$ So, the derivative of the given function is: $$\frac{d}{dx}\left(\ln(\cos^2 x)\right) = -2 \sin x$$

Key concepts

Derivatives Calculus

Understanding the concept of derivative is fundamental in calculus. Derivatives represent the rate at which a function is changing at any given point and are essential for solving problems involving motion, growth, and decay.

Imagine driving a car and watching the speedometer; the speed that you see is like the derivative of your position with respect to time. In the context of mathematics, if you have a function, let's say, the function describes the position of the car over time, the derivative of this function gives you the function's instantaneous rate of change, or how fast the car's position is changing at any point in time.

In calculus, derivatives are found using various rules. One of these rules is the chain rule, which is particularly useful when dealing with composite functions - functions made up by putting other functions together. The exercise in question involves applying the chain rule to find the derivative of a composite function involving a logarithm and trigonometric function. To get the correct answer, it's vital to understand each component of the function and how they interact with each other.

Natural Logarithm Derivatives

When it comes to the derivatives of logarithmic functions, particularly the natural logarithm, there is a specific rule to follow. The natural logarithm, denoted as \(ln(x)\), has a derivative that is an algebraic representation of its rate of change.

The derivative of the natural logarithm of a function \(u(x)\) is given by \(\frac{1}{u(x)}\cdot u'(x)\), where \(u'(x)\) is the derivative of \(u(x)\). This rule expresses that the rate of change of a natural logarithm depends inversely on the function itself. In essence, as the value of \(u(x)\) grows larger, the slope or steepness of the \(ln(u(x))\) flattens out, indicating a slower rate of change.

In the context of the given exercise, the function is the natural logarithm of \(\cos^2(x)\). So, according to the rule, the derivative begins with \(\frac{1}{\cos^2(x)}\), which represents the slope of the logarithm at any point \(x\). The next step involves finding the derivative of the inside function, \(\cos^2(x)\), which brings us to the derivatives of trigonometric functions.

Trigonometric Functions Derivatives

Trigonometric functions like sine, cosine, and tangent each have their own specific derivatives. For calculus students, these derivatives become essential tools in calculating rates of change in problems involving angles or periodic phenomena.

The derivative of \(\sin(x)\) is \(\cos(x)\), symbolizing that the rate of change of the sine function at any angle \(x\) is the cosine of that angle. Conversely, the derivative of \(\cos(x)\) is \(\-\sin(x)\), indicating a decreasing rate of change as \(\cos(x)\) progresses.

The exercise provided showcases the derivative of a squared cosine function, \(\cos^2(x)\). Finding this derivative involves understanding that \(\cos^2(x)\) is essentially \(\cos(x) \cdot \cos(x)\), and therefore requires the use of the product rule, which states that the derivative of two multiplied functions is the first times the derivative of the second plus the second times the derivative of the first. Simplifying this using the known derivatives of cosine, we arrive at the expression \(\-2 \cos(x) \sin(x)\), which reflects the rate of change for our squared trigonometric function.

By combining the calculated derivatives of the trigonometric inner function and the logarithmic outer function, students can fully solve the exercise at hand.