Question

What is the derivative of `log_10((x^2-1)/x)`?

Answer 1

see below

Explanation:

Use the following Properties of Logarithm to expand the problem before taking derivatives.

1. `color(red)(log_b(xy)=log_bx+log_by`
2. `color(red)(log_b(x/y)=log_bx-log_by`
3. `color(red)(log_b x^n =n log_bx`

Then use the formula `color(red)(d/dx(log_bf(x))=1/(f(x)ln b) * f'(x)` to find the derivative

`y=log_10((x^2-1)/x)`

`=log_10(x^2-1)-log_10 x`

`color(blue)(y'=1/((x^2-1)ln10)*2x-1/(xln10)`

`color(blue)(y'=(2x)/((x^2-1)ln10)-1/(xln10)`