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

Verified step by step guidance

1. Start by taking the natural logarithm of both sides of the function: ln(f(x)) = ln((1 + 1/x)^x).
2. Use the properties of logarithms to simplify the right side: ln(f(x)) = x * ln(1 + 1/x).
3. Differentiate both sides with respect to x using implicit differentiation: (1/f(x)) * f'(x) = ln(1 + 1/x) + x * (d/dx[ln(1 + 1/x)]).
4. Calculate the derivative of ln(1 + 1/x) using the chain rule: d/dx[ln(1 + 1/x)] = -1/(x(1 + 1/x)).
5. Substitute this derivative back into the equation and solve for f'(x) by multiplying both sides by f(x).