Problem 33 - Thomas Calculus

Question

In Exercises \(7-38,\) 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 is \( \frac{\tan(\ln \theta)}{\theta} \).

Step 1: Identify the Function Structure

Identify the function to differentiate, which is composed of the natural logarithm of the secant function that itself contains another natural logarithm. Hence, \( y = \ln(\sec(\ln \theta)) \).

Step 2: Apply Chain Rule

To find the derivative of this complex function, we apply the chain rule. This requires multiple derivative calculations for our nested functions, starting with the outermost function.

Step 3: Differentiate Outer Function

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

Step 4: Differentiate Secant Function

Next, differentiate the secant function \( \sec(v) \), where \( v = \ln \theta \). The derivative of \( \sec(v) \) with respect to \( v \) is \( \sec(v)\tan(v) \).

Step 5: Differentiate Innermost Function

Finally, differentiate the innermost function \( \ln \theta \) with respect to \( \theta \). This results in \( \frac{1}{\theta} \).

Step 6: Combine Derivatives with Chain Rule

Using the chain rule, combine all derivatives: The derivative of the outer function with respect to \( u \) is \( \frac{1}{\sec(\ln \theta)} \), the derivative of \( \sec(\ln \theta) \) with respect to \( \ln \theta \) is \( \sec(\ln \theta)\tan(\ln \theta) \), and the derivative of \( \ln \theta \) with respect to \( \theta \) is \( \frac{1}{\theta} \). The product is: \[ \frac{dy}{d\theta} = \frac{1}{\sec(\ln \theta)} \cdot \sec(\ln \theta) \tan(\ln \theta) \cdot \frac{1}{\theta} \].

Step 7: Simplify the Expression

Simplify the expression by canceling \( \sec(\ln \theta) \) from the numerator and denominator, yielding: \[ \frac{dy}{d\theta} = \frac{\tan(\ln \theta)}{\theta} \].

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus. It is particularly useful when dealing with composite functions, where one function is nested inside another. This technique allows us to differentiate complex, multi-layered functions by breaking them down into simpler parts. In essence, the chain rule tells us that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. It follows the formula:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Here's how it works:

• Identify the outer and inner functions.
• Differentiate the outer function while keeping the inner function unchanged.
• Differentiate the inner function.
• Multiply the derivatives together as prescribed by the chain rule.

In our exercise, we applied the chain rule to differentiate a function composed of a natural logarithm and a secant function, each with their own inner functions.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is the logarithm to the base \(e\), where \(e\) is approximately equal to 2.71828. It is an essential concept in calculus and mathematical analysis. The natural logarithm is powerful because it easily deals with exponential growth and decay phenomena.
When differentiating a natural logarithm, the derivative of \( \ln(x) \) is \( \frac{1}{x} \). This property holds true for any natural logarithm of a function \(u\), not just \(x\). Thus, if you have \( \ln(u) \), the derivative would be \( \frac{1}{u} \cdot \frac{du}{dx} \), applying the chain rule. In our problem, the outermost function was \(\ln(\sec(\ln \theta))\), so its derivative factored into the chain rule's structure.

Secant Function

The secant function, denoted as \( \sec(\theta) \), is one of the fundamental trigonometric functions, reciprocal to the cosine function. Its mathematical definition is given by \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
In differentiation, the derivative of \( \sec(v) \) with respect to \(v\) is \( \sec(v) \tan(v) \), where \(\tan(v)\) is the tangent of \(v\). This derivative formula arises from combining the rules for trigonometric derivatives and the quotient rule, because secant is a reciprocal function.
In our exercise, we applied this derivation to \( \sec(\ln \theta) \) when calculating the chain rule. Recognizing this differentiation pattern is crucial in handling trigonometric functions that are involved in composite expressions.

Differentiation Steps

Differentiation steps help systematically break down the process of finding the derivative of complex expressions. The original problem involved multiple layers where each layer required analysis and careful differentiation.

• **Step 1:** Recognize the function's structure, identify the layers: \( \ln \), \( \sec \), and another \( \ln \).
• **Step 2:** Apply the chain rule, starting from the outermost function. Differentiate step-by-step inward.
• **Step 3:** Differentiate \( \ln(u) \), then \( \sec(v) \) and finally inner \( \ln \theta \).
• **Step 4:** Combine derivatives using the chain rule, and multiply them sequentially.
• **Step 5:** Simplify the expression, cancel common terms like \( \sec(\ln \theta) \).

This systematic approach makes differentiation manageable and reduces errors. Each step builds on the last, ensuring the final derivative is correct. In our exercise, after cancelling terms, the final simplified derivative is \( \frac{\tan(\ln \theta)}{\theta} \). Understanding these steps well enables you to tackle any function with confidence.