Problem 62 - Thomas Calculus in SI Units

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 Differentiation Rule

In this problem, we need to differentiate an expression of the form \(y = (u)^{a}\), where \(u = \ln \theta\) and \(a = \pi\). Here, \(a\) is a constant, so we will use the Power Rule and the Chain Rule to differentiate.

Step 2: Apply the Power Rule

The Power Rule for derivatives states that if \(y = u^a\), then \(\frac{dy}{du} = a \cdot u^{a-1}\). Applying this to our function, we have \[ \frac{dy}{d(\ln \theta)} = \pi \cdot (\ln \theta)^{\pi-1}. \]

Step 3: Differentiate the Inner Function

Next, apply the Chain Rule, which requires differentiating the inner function \(u = \ln \theta\) with respect to \(\theta\). The derivative is \[ \frac{du}{d\theta} = \frac{1}{\theta}. \]

Step 4: Apply the Chain Rule

Combine the results from Step 2 and Step 3 using the Chain Rule, which states that \[ \frac{dy}{d\theta} = \frac{dy}{d(\ln \theta)} \cdot \frac{d(\ln \theta)}{d\theta}. \] Substituting, we get \[ \frac{dy}{d\theta} = \pi \cdot (\ln \theta)^{\pi-1} \cdot \frac{1}{\theta}. \]

Step 5: Simplify the Expression

Simplify the derivative obtained: \[ \frac{dy}{d\theta} = \frac{\pi (\ln \theta)^{\pi-1}}{\theta}. \] This is the simplified form of the derivative of \(y\) with respect to \(\theta\).

Key Concepts

Power Rule

When it comes to finding the derivative of functions like powers of variables, the Power Rule is a fundamental calculus tool. It's straightforward and efficient for differentiating expressions in the form of \(y = u^a\), where \(u\) is a function of some variable, and \(a\) is a constant. In this case, the Power Rule gives us a method to find the derivative by applying the formula: \[ \frac{dy}{du} = a \, u^{a-1} \].This rule effectively tells us how a slight change in \(u\) affects \(y\). In our example, this means taking the exponent \(\pi\) and multiplying it by \((\ln \theta)^{\pi-1}\).
From a practical perspective:

• The constant \(a\) becomes a coefficient.
• The exponent is reduced by one.
• The function's new power is \(a-1\).

This step essentially handles the outer layer of our composite function.

Chain Rule

The Chain Rule is a powerful technique used when dealing with composite functions. It allows us to differentiate "layers" of functions by breaking them down into simpler, more manageable parts. In calculus, whenever you encounter a scenario where one function is nested inside another, the Chain Rule becomes your go-to method.
The rule states that to find the derivative of \(y = f(g(x))\), we take the derivative of the outer function \(f\) with respect to \(g\), and multiply it by the derivative of the inner function \(g\) with respect to \(x\). This can be represented as:\[ \frac{dy}{dx} = \frac{dy}{dg} \times \frac{dg}{dx} \].In our problem, the function \(u = \ln \theta\) is inside another function. We've dealt with the outer layer using the Power Rule, and then the Chain Rule instructs us to find the derivative of this inner function too.

• Differentiate the outer function (already applied in the Power Rule).
• Differentiate the inner function \(\ln \theta\) as \(\frac{1}{\theta}\).
• Multiply the results to get the complete derivative.

Each function layer has its derivative considered, then they're multiplied, creating a link in the derivative chain.

Natural Logarithm

The natural logarithm, represented as \(\ln(x)\), is a special function in mathematics, linked closely to natural growth processes and logarithmic scales. In calculus, understanding how \(\ln(x)\) behaves when differentiated is essential. Here's a concise breakdown:
The derivative of \(\ln(x)\) follows a clear and consistent rule: \[ \frac{d}{dx}(\ln(x)) = \frac{1}{x} \]. This derivative emerges from the unique properties of the natural logarithm and serves to simplify the process of finding slopes of log functions.

• It's only applicable for \(x > 0\) as the natural log is undefined for zero or negative values.
• The function \(\ln(x)\) grows steadily, meaning its derivative reduces as \(x\) increases.
• For inverse functions, it showcases logarithmic decrease.

In our exercise, \(\ln \theta\) forms the inner function. By knowing its derivative, \(\frac{1}{\theta}\), we can effectively apply this in tandem with the Power and Chain Rules to fully simplify our derivative expression of the overall function. Comprehending \(\ln(x)\) allows us to untangle such equations seamlessly.