75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = x^In x

Verified step by step guidance

1. Step 1: Start by taking the natural logarithm of both sides of the equation y = x^(ln x). This gives us ln(y) = ln(x^(ln x)).
2. Step 2: Use the property of logarithms that allows you to bring the exponent down: ln(y) = ln x * ln x.
3. Step 3: Differentiate both sides with respect to x. For the left side, use implicit differentiation: (1/y) * dy/dx. For the right side, use the product rule: d(ln x * ln x)/dx = ln x * (1/x) + ln x * (1/x).
4. Step 4: Solve for dy/dx by multiplying both sides by y: dy/dx = y * (ln x/x + ln x/x).
5. Step 5: Substitute back y = x^(ln x) to express dy/dx in terms of x: dy/dx = x^(ln x) * (2 ln x/x).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products or quotients of variables raised to variable powers. By taking the natural logarithm of both sides of the function, we can simplify the differentiation process, especially when dealing with complex expressions. This method is particularly useful for functions like f(x) = x^ln(x), where both the base and the exponent are variable.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential when applying logarithmic differentiation, as it allows us to handle the derivatives of the inner and outer functions effectively.

Exponential and Logarithmic Functions

Exponential and logarithmic functions are inverses of each other, with exponential functions having the form y = a^x and logarithmic functions expressed as x = log_a(y). Understanding their properties is crucial for logarithmic differentiation, as it involves manipulating these functions to simplify the differentiation process. For instance, knowing that ln(a^b) = b*ln(a) helps in breaking down the function f(x) = x^ln(x) into manageable parts.