Question

Use the properties of logarithms to simplify the following functions before computing \(f^{\prime}(x)\). $$f(x)=\ln \left(\sec ^{4} x \tan ^{2} x\right)$$

Short answer

Question: Find the derivative of the function \(f(x) = \ln(\sec^4x \tan^2x)\). Answer: The derivative of the function is \(f^{\prime}(x) = -2\cos x\tan x\sec^2x\).

Step-by-step solution

Simplify the function using logarithm properties

We are given the function \(f(x)=\ln \left(\sec ^{4} x \tan ^{2} x\right)\). Recall that \(\ln(ab) = \ln(a) + \ln(b)\). Apply this property to the given function: $$f(x) = \ln\left(\sec^4x\right) + \ln\left(\tan^2x\right)$$ Also, remember that \(\ln(a^b) = b \cdot \ln(a)\). Utilizing this property, we get: $$f(x) = 4\ln\left(\sec x\right) + 2\ln\left(\tan x\right)$$

Recall trigonometric identity

Now recall the trigonometric identity \(\sec x = \frac{1}{\cos x}\) and rewrite the expression for f(x): $$f(x) = 4\ln\left(\frac{1}{\cos x}\right) + 2\ln\left(\tan x\right)$$

Find the derivative of the simplified function

Now differentiate the simplified function with respect to x using the chain rule: $$f^{\prime}(x) = \frac{d}{dx}\left[4\ln\left(\frac{1}{\cos x}\right) + 2\ln\left(\tan x\right)\right]$$ Chain rule: \(\frac{d}{dx}\ln(u(x)) = \frac{u^{\prime}(x)}{u(x)}\). Apply the chain rule to differentiate f(x): $$f^{\prime}(x)= 4\frac{-\sin x}{\cos^2x} + 2\frac{\sec^2x}{\tan x}$$

Rewrite the derivative using trigonometric identities

Recall that \(\tan x = \frac{\sin x}{\cos x}\) and \(\sec^2x = \frac{1}{\cos^2x}\). Rewrite the derivative using these identities: $$f^{\prime}(x)= 4\frac{-\sin x}{\cos^2x} + 2\frac{1}{\cos^2x}\cdot\frac{\sin x}{\cos x}$$

Simplify the derivative

Factor out the common term \(\frac{1}{\cos^2x}\): $$f^{\prime}(x)= \frac{1}{\cos^2x}\left(-4\sin x + 2\sin x\right)$$ Simplify the expression inside the parenthesis: $$f^{\prime}(x)= \frac{1}{\cos^2x}\left(-2\sin x\right)$$ Finally, to write the answer in terms of trigonometric functions, we can use the identity \(\sin x = \cos x \cdot \tan x\) to obtain: $$f^{\prime}(x)= -2\sin x\sec^2x = -2\cos x\tan x\sec^2x$$

Key concepts

Chain Rule Differentiation

Understanding the chain rule is essential when dealing with complex functions, particularly when differentiating compositions of functions. In essence, the chain rule tells us how to find the derivative of a function of a function. The classic form of the chain rule is expressed as \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \).

Let's apply this to our context using the logarithmic function. Suppose we have a function \( u(x) \) whose derivative is \( u'(x) \) and we want to find the derivative of \( \ln(u(x)) \). Using the chain rule, \( \frac{d}{dx}\ln(u(x)) = \frac{u'(x)}{u(x)} \), because the derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{1}{x} \). This process is what allows us to move through the steps of differentiation in the original exercise with accuracy and confidence.

By acknowledging and applying the chain rule correctly, we are able to handle the derivatives of complex expressions efficiently, breaking them down into more manageable parts.

Trigonometric Identities

In calculus and particularly when working with trigonometric functions, knowing your trigonometric identities is indispensable. These identities are equalities involving trigonometric functions that hold for all values of the variable where both sides of the equality are defined.

Commonly used identities include \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \) among others. These identities help us to transform and simplify trigonometric expressions. For instance, in the exercise, the identity \( \sec x = \frac{1}{\cos x} \) is used to rewrite \( \ln(\sec^4x) \) as \( 4\ln(\frac{1}{\cos x}) \).

Understanding and memorizing these identities are key to working through derivatives and integrals of trigonometric functions, allowing one to rewrite functions in a more 'calculus-friendly' format.

Derivative of Logarithmic Functions

Taking the derivative of logarithmic functions might seem daunting at first, but it follows a direct and consistent rule. For the natural logarithm \( \ln(x) \), the derivative is \( \frac{d}{dx}\ln(x) = \frac{1}{x} \). But, what if \( x \) itself is a more complex function, say \( u(x) \) as discussed in the chain rule? In that case, the derivative becomes \( \frac{d}{dx}\ln(u(x)) = \frac{u'(x)}{u(x)} \).

During the exercise, the function \( \ln(\sec^4 x \tan^2 x) \) is initially a complex expression, but after utilizing properties of logarithms and rewriting it in terms of \( \sec x \) and \( \tan x \) separately, differentiation becomes much more manageable. Recognizing this process is crucial for solving similar problems, and the derivative of the logarithmic function serves as the backbone for the solution.

Simplifying Logarithmic Expressions

When faced with simplifying logarithmic expressions, one must draw upon specific properties of logarithms to break down intricate expressions into more solvable pieces. Among the many properties, two are particularly useful: \( \ln(ab) = \ln(a) + \ln(b) \) and \( \ln(a^b) = b \cdot \ln(a) \).

In the problem provided, the initial expression \( \ln(\sec^4 x \tan^2 x) \) is simplified by using these properties. First, by separating the logarithm of a product into the sum of logarithms, and then by moving the exponent in front of the logarithm as a coefficient. By simplifying the logarithmic expression before differentiating, the overall task becomes much less complex, allowing us to apply the chain rule and trigonometric identities effectively. Recognizing and applying these properties can greatly reduce the complexity of calculus problems involving logarithms.