Problem 26 - Thomas Calculus

Question

In Exercises \(17-36,\) find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$ y=\ln \left(3 \theta e^{-\theta}\right) $$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{d\theta} = \frac{1}{\theta} - 1 \).

Step 1: Identify the Differentiation Rule Needed

The function given is a logarithmic function: \( y = \ln(3\theta e^{-\theta}) \). To differentiate this, we will apply the chain rule and the properties of logarithms.

Step 2: Simplify the Logarithmic Expression

Use the logarithmic identity \( \ln(ab) = \ln a + \ln b \) to simplify the expression: \[ y = \ln(3\theta e^{-\theta}) = \ln 3 + \ln \theta + \ln e^{-\theta}. \] Then, simplify further using \( \ln e^{-\theta} = -\theta. \) This gives: \[ y = \ln 3 + \ln \theta - \theta. \]

Step 3: Differentiate Each Term

Differentiate each term separately:- The derivative of \( \ln 3 \) is 0, since it is a constant.- The derivative of \( \ln \theta \) with respect to \( \theta \) is \( \frac{1}{\theta} \).- The derivative of \( -\theta \) is \( -1 \).

Step 4: Combine All Parts to Find the Derivative

Combine the derivatives from each term in Step 3: \[ \frac{dy}{d\theta} = 0 + \frac{1}{\theta} - 1. \] Thus, the derivative is: \[ \frac{dy}{d\theta} = \frac{1}{\theta} - 1. \]

Key Concepts

Chain Rule

The chain rule is one of the foundational techniques in calculus, used to differentiate composite functions. When you have a function nested within another function, such as the case with the function \( y = \ln(3\theta e^{-\theta}) \), the chain rule comes into play. It enables us to tackle the differentiation step by step, working from the outside inwards.

• First, identify the outer function and the inner function.
• Differentiate the outer function with respect to the inner function.
• Then multiply by the derivative of the inner function with respect to its own variable.

In the example provided, our base function is the natural logarithm, and our inner function is the expression \( 3\theta e^{-\theta} \). Understanding the chain rule simplifies the process of differentiation considerably, especially for complex functions involving multiple layers.

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique used to differentiate functions that involve products or quotients, which can be cumbersome if expanded directly. The function \( y = \ln(3\theta e^{-\theta}) \) showcases the necessity and convenience of using properties of logarithms during differentiation. Here's how it works:

• Use logarithm identities, such as \( \ln(ab) = \ln a + \ln b \).
• This simplifies multiplicative complexities into additive easier-to-handle parts.

For example, by applying the property to \( \ln(3\theta e^{-\theta}) \), we break it down into \( \ln 3 + \ln \theta + \ln e^{-\theta} \). This simplification streamlines differentiation because each term can be handled one at a time, avoiding potential errors and saving time.

Differentiation Techniques

Differentiation techniques encompass various rules and methods used to find the derivative of functions. When addressing a problem, such as the one with \( y = \ln(3\theta e^{-\theta}) \), it's essential to recognize which techniques are applicable. The most common differentiation rules include:

• Power Rule: Used for functions of the form \( x^n \).
• Product Rule: Applies when differentiating products of two functions.
• Quotient Rule: Used when you need to differentiate a quotient of two functions.
• Chain Rule: Essential for composite functions, as discussed previously.
• Logarithmic Differentiation: Particularly helpful for complex products or quotients.

By understanding and selecting the correct technique, such as logarithmic differentiation and the chain rule in our example, we navigate the differentiation process more effectively. This not only leads to a correct solution but also improves our problem-solving skills in calculus, making us better equipped to tackle various types of functions.