Problem 18 - Thomas Calculus

Question

Find the derivative of \(y\) with respect to the given independent variable. \(y=(\ln \theta)^{\pi}\)

Step-by-Step Solution

Answer

The derivative is \(\frac{\pi (\ln \theta)^{\pi - 1}}{\theta}\).

Step 1: Identify the Function Form

The function given is in the form of an exponentiation: \[y = (\ln \theta)^\pi\]This suggests the use of differentiation techniques suitable for exponential functions, particularly the chain rule and the power rule.

Step 2: Apply the Chain Rule

To differentiate \((\ln \theta)^\pi\), express it in a form that allows application of the chain rule:Set \[y = u^\pi, \text{ where } u = \ln \theta\].The derivative of \(y\) with respect to \(\theta\) is:\[\frac{dy}{d\theta} = \frac{dy}{du} \cdot \frac{du}{d\theta}\].

Step 3: Differentiate Using the Power Rule

First, differentiate \(y = u^\pi\) with respect to \(u\):\[\frac{dy}{du} = \pi u^{\pi-1}\].

Step 4: Differentiate the Inner Function

Differentiate \(u = \ln \theta\) with respect to \(\theta\):\[\frac{du}{d\theta} = \frac{1}{\theta}\].

Step 5: Combine Using the Chain Rule

Substitute the derivatives back into the chain rule formula:\[\frac{dy}{d\theta} = \pi (\ln \theta)^{\pi - 1} \cdot \frac{1}{\theta} = \frac{\pi (\ln \theta)^{\pi - 1}}{\theta}\].

Key Concepts

Understanding the Chain Rule

When faced with differentiating a composite function, the chain rule becomes a valuable tool in calculus. The chain rule states 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. This is particularly useful when dealing with functions inside other functions. For example, if you have a function like

• \( y = (f(g(x))) \)
• You first find \( \frac{dy}{dg} \), the derivative of the outer function \( f \) with respect to \( g \).
• Then find \( \frac{dg}{dx} \), the derivative of the inner function \( g \) with respect to \( x \).

The overall derivative is \[\frac{dy}{dx} = \frac{dy}{dg} \cdot \frac{dg}{dx}\]This allows us to break down complex functions into more manageable parts. In practice, this technique helps simplify the differentiation of functions that involve multiple expressions nested within each other.

Exploring Exponential Functions

Exponential functions have a constant raised to a variable power, and their unique property is that their rate of growth is proportional to their current value. However, in calculus, we sometimes encounter variables raised to fixed powers, like

• \( y = u^\pi \)

It still calls for differentiation techniques similar to exponential functions but requires attention to the base being variable.
For functions like \( (\ln \theta)^\pi \), \( \ln \theta \) acts as the base, and the function involves transforming it into a power form. Such situations often require combining exponential function logic with other calculus rules to find the derivative effectively. Distinguishing between traditional exponential forms and functions where the base or the exponent varies is key.

Mastering the Power Rule

The power rule is one of the simplest differentiation techniques, used widely in calculus. It is applied when you are differentiating a term of the form

• \( x^n \)
• Here, \( n \) is any real number.

To differentiate \( x^n \), you bring the exponent \( n \) down in front of the variable and then subtract one from the exponent:\[\frac{d}{dx}(x^n) = n \cdot x^{n-1}\]In our original exercise, this rule is combined with the chain rule. When differentiating

• \( y = (\ln \theta)^\pi \)
• We treat \( (\ln \theta) \) as our base \( u \), allowing the power rule's application.

Upon differentiating \( u^\pi \), the power rule helps format the expression into a simpler function through which you can easily manage and work through more complex calculus problems.