State the derivative rule for the logarithmic function f(x)=log(subscript b)x. How does it differ from the derivative formula for ln x?

Verified step by step guidance

1. Recall that the derivative of the logarithmic function f(x) = log_b(x) can be expressed using the change of base formula: log_b(x) = ln(x) / ln(b).
2. Differentiate f(x) = log_b(x) using the quotient rule, which states that if you have a function h(x) = u(x)/v(x), then h'(x) = (u'v - uv')/v^2.
3. In this case, let u(x) = ln(x) and v(x) = ln(b). The derivative of u(x) is u'(x) = 1/x, and since v(x) = ln(b) is a constant, v'(x) = 0.
4. Substituting these derivatives into the quotient rule gives f'(x) = (1/x * ln(b) - ln(x) * 0) / (ln(b))^2 = 1 / (x * ln(b)).
5. In contrast, the derivative of ln(x) is simply 1/x, which shows that the presence of the base b in log_b(x) introduces the factor of 1/ln(b) in the derivative.