Question

Найдите производную y=log x^2 по основанию (х-1)

Answer 1

Перейдем к новому основанию (е)
$log _{(x-1)} x^2= \frac{ln x^{2} }{ln(x-1)} \\ \\ y'=( \frac{ln x^{2} }{ln(x-1)})'= \frac{(ln x^{2} )'ln(x-1)-(ln(x-1))'ln x^{2} }{ln ^{2} (x-1)} = \frac{ \frac{2ln(x-1)}{x} - \frac{ln x^{2} }{x-1} }{ln ^{2} (x-1)} = \\ \\ = \frac{2ln(x-1)}{xln ^{2} (x-1)} - \frac{ln x^{2} }{(x-1)ln ^{2} (x-1)}= \frac{2}{xln (x-1)} - \frac{ln x^{2} }{(x-1)ln ^{2} (x-1)}$