Problem 45 - Thomas Calculus

Question

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. \(y=x^{\ln x}\)

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = 2x^{\ln x - 1} \ln x \).

Step 1: Take Natural Logarithm of Both Sides

Start with the original function: \( y = x^{\ln x} \). Apply the natural logarithm to both sides to transform the power function. This gives us: \( \ln y = \ln(x^{\ln x}) \). By applying the logarithm power rule, simplify the right side to get: \( \ln y = (\ln x) \cdot \ln x \) or \( (\ln x)^2 \).

Step 2: Differentiate Implicitly with Respect to x

Differentiate both sides of the equation \( \ln y = (\ln x)^2 \) with respect to \( x \). The left side, \( \ln y \), becomes \( \frac{1}{y} \cdot \frac{dy}{dx} \) using the chain rule. The right side, \( (\ln x)^2 \), differentiates to \( 2\ln x \cdot \frac{1}{x} \) using the product and chain rules. Therefore, the equation is \( \frac{1}{y} \cdot \frac{dy}{dx} = \frac{2\ln x}{x} \).

Step 3: Solve for \(\frac{dy}{dx}\)

Multiply both sides by \( y \) to isolate \( \frac{dy}{dx} \) and find the derivative: \( \frac{dy}{dx} = y \cdot \frac{2\ln x}{x} \).

Step 4: Substitute Back for y

Recall that \( y = x^{\ln x} \). Substitute this back into the derivative equation: \( \frac{dy}{dx} = x^{\ln x} \cdot \frac{2\ln x}{x} \). Simplify this to get the derivative in its final form: \( \frac{dy}{dx} = 2x^{\ln x - 1} \ln x \).

Key Concepts

Natural Logarithm

The natural logarithm, often denoted by \( \ln \), is a special type of logarithm. It's based on the constant \( e \), which is approximately 2.71828. The natural logarithm helps us deal with exponential functions efficiently, especially when they are complex or involve variables in exponents.

In the exercise given, we start with the expression \( y = x^{\ln x} \). This is a power function where the exponent itself includes a logarithmic function. To make differentiating manageable, we apply the natural logarithm to both sides, transforming it into \( \ln y = \ln(x^{\ln x}) \).

• The property \( \ln(a^b) = b \cdot \ln a \) simplifies the expression. This step makes differentiation feasible by turning a complex power function into a simple product.

Implicit Differentiation

Implicit differentiation is a technique used when dealing with functions that aren’t isolated, or explicitly stated, like \( y = f(x) \). This is particularly useful when both sides of an equation involve the dependent variable \( y \) and the independent variable \( x \).

In our task, after applying the natural logarithm, we get \( \ln y = (\ln x)^2 \), which involves an implicit form because \( y \) isn't isolated. To find the derivative \( \frac{dy}{dx} \), we differentiate both sides with respect to \( x \).

• On the left, we apply the derivative of \( \ln y \), which becomes \( \frac{1}{y} \cdot \frac{dy}{dx} \), using the chain rule.
• The right side differentiates using the power rule for the natural log function, producing \( 2 \ln x \cdot \frac{1}{x} \).

Implicit differentiation allows us to handle this complex relationship and find the derivative accurately.

Chain Rule

The chain rule is a fundamental calculus principle for finding the derivative of composite functions. It states that \( \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \). This rule helps manage differentiation when a function is embedded within another.

During our differentiation process, the chain rule is essential. For instance, differentiating \( \ln y \) with respect to \( x \), where \( y \) itself depends on \( x \), requires chaining the processes:

• We consider the external function \( \ln y \), whose derivative concerning \( y \) is \( \frac{1}{y} \).
• This is "chained" with the derivative of \( y \) with respect to \( x \), which is \( \frac{dy}{dx} \).

Similarly, on the right side of the equation, the compound structure \( (\ln x)^2 \) uses the chain rule when differentiating to become \( 2 \ln x \cdot \frac{1}{x} \). Understanding and applying the chain rule enables the correct and efficient computation of derivatives of nested functions.

Logarithm Power Rule

The logarithm power rule is a property that simplifies expressions involving logarithmic functions, especially when an exponent is present. It states that \( \ln(a^b) = b \cdot \ln a \). This rule is based on the properties of exponents and logarithms working in reverse operations.

In the provided exercise, we faced \( \ln(x^{\ln x}) \), where the power rule transforms it into \( (\ln x) \cdot \ln x \) or equivalently \( (\ln x)^2 \).

• This transformation is crucial for simplifying the differentiation process.
• The rule allows breaking down complex expressions into simpler factors that can be individually differentiated.

Essentially, the logarithm power rule gracefully handles the exponentiation within logarithmic expressions, providing a clear path towards differentiation.