75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = (1+ 1/x)^x

Verified step by step guidance

1. Step 1: Begin by taking the natural logarithm of both sides of the equation. This gives you ln(f(x)) = ln((1 + 1/x)^x).
2. Step 2: Use the property of logarithms that allows you to bring the exponent down: ln(f(x)) = x * ln(1 + 1/x).
3. Step 3: Differentiate both sides with respect to x. On the left side, use implicit differentiation: d/dx[ln(f(x))] = (1/f(x)) * f'(x). On the right side, apply the product rule to differentiate x * ln(1 + 1/x).
4. Step 4: For the right side, differentiate x * ln(1 + 1/x) using the product rule: d/dx[x * ln(1 + 1/x)] = ln(1 + 1/x) + x * (1/(1 + 1/x)) * (-1/x^2).
5. Step 5: Solve for f'(x) by multiplying both sides by f(x): f'(x) = f(x) * [ln(1 + 1/x) - 1/(x * (1 + 1/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 exponential forms. This method is particularly useful for functions like f(x) = (1 + 1/x)^x, where direct differentiation can be cumbersome.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y = g(u) and u = f(x), then the derivative dy/dx can be found by multiplying the derivative of g with respect to u by the derivative of f with respect to x. This rule is essential when applying logarithmic differentiation, as it allows us to differentiate the logarithm of a function effectively.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a constant and 'x' is a variable. These functions exhibit rapid growth or decay and are characterized by their unique property that the rate of change is proportional to the function's value. Understanding the behavior of exponential functions is crucial when evaluating derivatives of functions like f(x) = (1 + 1/x)^x, as they often involve limits and asymptotic analysis.