Problem 14 - Thomas Calculus in SI Units

Question

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 Derivative Method

We need to find the derivative of the function \(y = \ln(3 \theta e^{- heta})\) with respect to \(\theta\). This function is a natural logarithm applied to a product, so we can use the properties of logarithms and the chain rule for differentiation.

Step 2: Simplify the Expression Inside the Logarithm

Use the property \(\ln(ab) = \ln a + \ln b\) to simplify the expression: \(y = \ln(3\theta e^{-\theta}) = \ln 3 + \ln \theta + \ln(e^{-\theta})\).

Step 3: Differentiate Each Term Separately

Now that we have \(y = \ln 3 + \ln \theta - \theta\), differentiate each term with respect to \(\theta\):1. The derivative of \(\ln 3\), a constant, is 0.2. The derivative of \(\ln \theta\) is \(\frac{1}{\theta}\). 3. The derivative of \(-\theta\) is \(-1\).

Step 4: Compile the Final Derivative

Combine the derivatives of each term: \(\frac{dy}{d\theta} = 0 + \frac{1}{\theta} - 1 = \frac{1}{\theta} - 1\).

Key Concepts

Derivative

In calculus, the derivative represents the rate at which a function is changing at any given point. It's a fundamental tool for analyzing mathematical models in physics, economics, engineering, and many other fields.

The derivative at a point on a function can be conceptualized as the slope of the tangent line to the curve at that point. For a function \(y=f(x)\), the derivative is often denoted as \(\frac{dy}{dx}\), which literally means "the change in \(y\) with respect to \(x\)."

When finding the derivative of a function, you can use various rules and techniques, such as the power rule, product rule, and quotient rule. In this exercise, we focus on the chain rule and logarithmic differentiation as the key methods to simplify and compute the derivative of a natural logarithm of a product.

Chain Rule

The chain rule is a powerful method used in calculus for finding the derivative of composite functions. A composite function is a function composed of two or more functions, where one function is applied to the result of another function.

Mathematically, if you have a composite function \(y = f(g(x))\), the chain rule states that the derivative of \(y\) with respect to \(x\) is given by:

• \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\)

In simple terms, you take the derivative of the outer function evaluated at the inner function, and multiply it by the derivative of the inner function.

In the exercise provided, the chain rule plays a role in handling the exponential function inside the logarithm. By understanding and applying the chain rule, the differentiation process becomes more manageable, especially when dealing with complex expressions.

Logarithmic Differentiation

Logarithmic differentiation is an alternative method for differentiating functions, particularly useful when dealing with products or quotients of functions that are cumbersome to handle with the standard rules. This technique leverages the properties of logarithms to simplify differentiation.

The key property here is the logarithm of a product, which is the sum of the logarithms: \(\ln(ab) = \ln a + \ln b\). Applying this property allows us to break down a complex product into a sum of simpler terms, making it easier to differentiate. Similarly, the logarithm of a quotient can be expressed as the difference of logarithms: \(\ln \left(\frac{a}{b}\right) = \ln a - \ln b\).

In the original problem, the function within the logarithm \(3\theta e^{-\theta}\) was simplified to \(\ln 3 + \ln \theta - \theta\) before differentiating. Each term could then be differentiated individually, making the overall process simpler and more intuitive. Using logarithmic differentiation is particularly powerful when you find yourself dealing with products or powers that can otherwise be challenging with direct differentiation methods.