Question

What is the derivative of `log_3((xsqrt(x-1))/2)`?

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`

Use the formula `color(red)(d/dx(log_bx)=1/(xln b)` to find the derivative

`y=log_3((xsqrt(x-1))/2)=log_3((x(x-1)^(1/2))/2)`

`y=log_3x+log_3(x-1)^(1/2)-log_3 2`

`=log_3x+1/2 log_3(x-1)-log_3 2`

`color(blue)(y'=1/(xln3)+1/2 *1/(x-1)-0`

`color(blue)(y'=1/(xln3)+1/(2x-2)`