Calculus Math Help, please

Question

1. Why might one prefer to use the logarithmic differentiation for finding derivatives of functions that have another function as an exponent?

Answer 1

If you have a function such as h(x) = f(x)^g(x), taking the log of both sides allows the derivative product rule to be used.

ln(h(x)) = ln(f(x)^g(x)) = g(x)ln(f(x))

h'(x)/h(x) = g(x)f'(x)/f(x) +g'(x)ln(f(x))

h'(x) = h(x)[g(x)f'(x)/f(x) +g'(x)ln(f(x)]

= f(x)^g(x)[g(x)f'(x)/f(x) +g'(x)ln(f(x)]