Question

How do you find the derivative of `y = log_3 [(x+1/x-1)^ln3]`?

Answer 1

see below

Explanation:

Use the Property `color(red)(log_b x^n =n log_bx` to simplify the problem first and then use the the formula `color(red)(d/dx(log_b f(x))=1/(f(x)ln b) * f'(x)` to find the derivative

`y=log_3[(x+1/x-1)^ln3]`

`y=ln3* log_3(x+1/x-1)`

`color(blue)(y'=ln 3* 1/((x+1/x -1)ln 3)*(1-1/x^2)`

`color(blue)(y'=cancel ln 3* 1/((x+1/x -1)cancel ln 3)*(1-1/x^2)`

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

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

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

`color(blue)(y'= cancelx/(x^2-x+1) *(x^2-1)/(x^cancel2)`

`color(blue)(y'= (x^2-1)/(x^3-x^2+x)`