Problem 37 - University Calculus: Early Transcendentals

Question

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln (\sec (\ln \theta))$$

Step-by-Step Solution

Answer

The derivative of \(y\) with respect to \(\theta\) is \( \frac{\tan(\ln \theta)}{\theta} \).

Step 1: Identify the Composition of Functions

The function given is a composition, where we have \(y = \ln(\sec(\ln \theta))\). This means we need to differentiate using the chain rule multiple times, considering the innermost function first.

Step 2: Apply the Outer Derivative

The outermost function is the natural logarithm \(\ln(u)\) with \(u = \sec(\ln \theta)\). The derivative of \(\ln(u)\) is \(\frac{1}{u}\). So, the derivative of \(\ln(\sec(\ln \theta))\) with respect to \(u\) is \(\frac{1}{\sec(\ln \theta)}\).

Step 3: Differentiate the Middle Function

Now, consider \(u = \sec(v)\) where \(v = \ln \theta\). The derivative of \(\sec(v)\) with respect to \(v\) is \(\sec(v)\tan(v)\). Substitute back \(v = \ln \theta\) to get \(\sec(\ln \theta)\tan(\ln \theta)\).

Step 4: Differentiate the Innermost Function

The innermost function is \(v = \ln \theta\). The derivative of \(\ln \theta\) with respect to \(\theta\) is \(\frac{1}{\theta}\).

Step 5: Combine Using the Chain Rule

Now we apply the chain rule step-by-step:1. From Step 2: \(\frac{1}{\sec(\ln \theta)}\).2. Multiply by the derivative from Step 3: \(\sec(\ln \theta)\tan(\ln \theta)\).3. Multiply this by the derivative from Step 4: \(\frac{1}{\theta}\).This gives the final derivative: \( \frac{\sec(\ln \theta)\tan(\ln \theta)}{\sec(\ln \theta)\theta} = \frac{\tan(\ln \theta)}{\theta}\).

Key Concepts

Derivative

The derivative is a fundamental concept in calculus that helps us understand how a function changes as its input changes. When working with derivatives, you are essentially calculating the "slope" or rate of change of a function at any given point. In our problem, we are asked to find the derivative of a complex function involving y = \ln(\sec(\ln \theta)).
The task involves differentiating this function with respect to \(\theta\). To do this, we employ the chain rule, a powerful technique used for differentiating compositions of functions. By systematically breaking down the function into manageable parts, starting with the outermost function all the way to the innermost function, the chain rule allows us to compute the derivative methodically. This means:

• First, identifying and differentiating the outer function.
• Second, working through the middle function.
• Lastly, handling the innermost part and then combining the results.

With these steps, you can find the final result, which is the derivative of the entire function. The result here is \(\frac{\tan(\ln \theta)}{\theta}\). This pattern of differentiation showcases how derivatives not only apply to simple functions but also allow us to handle complex, layered functions by simplifying them step-by-step.

Natural logarithm

The natural logarithm, often denoted as \(\ln(x)\), is an essential mathematical function that helps model exponential growth and decay processes. It is the inverse of the exponential function \(e^x\). In the context of our problem, the natural logarithm appears multiple times in the function \(y = \ln(\sec(\ln \theta))\), making it a key component to address when we differentiate.
To differentiate a function that includes a natural logarithm, such as \(\ln(u)\), we use the formula \(\frac{d}{du}[\ln(u)] = \frac{1}{u}\). This calculation essentially gives us the rate at which the natural logarithm function changes as its input \(u\) changes.
In our exercise, we initially consider \(u = \sec(\ln \theta)\), which is a complicated composition in itself. By following the chain rule and systematically breaking down the natural logarithm function through its nested components, we reach the derivative \(\frac{1}{u}\) and continue the differentiation process into the deeper layers. Understanding how natural logarithms interact with other layers of functions is crucial for dealing with advanced calculus problems like this one.

Trigonometric functions

Trigonometric functions are indispensable tools in both calculus and the broader field of mathematics. In this exercise, we encounter the secant function, \(\sec(x)\), which is one of the six trigonometric functions. Traditional trigonometric functions include

• sine \(\sin\),
• cosine \(\cos\),
• tangent \(\tan\),
• and their reciprocals: cosecant \(\csc\), secant \(\sec\), and cotangent \(\cot\).

The secant function is defined as \(\sec(x) = \frac{1}{\cos(x)}\).
To find its derivative, we use the well-known formula \(\frac{d}{dx}[\sec(x)] = \sec(x)\tan(x)\). This derivative provides us the rate at which the secant changes as its input changes. In our particular problem, \(\theta\) is involved in two trigonometric-related transformations: first within the secant function, and also, the secant itself becomes a part of the function inside the logarithm.
By applying the chain rule, we ensure that each trigonometric function is properly differentiated in its sequence within the nested structure of our original function. Overall, this highlights the chain rule's effectiveness in tackling complexities involving both logarithmic and trigonometric layers.