Question

How do you find the derivative of `f(x)=ln(x^5(x-2)^3)`?

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`

That is,

`f(x)=ln (x^5 (x-2)^3)`

`=ln x^5 +ln (x-2)^3`

`=5 ln x+3 ln(x-2)`

`color(blue)(f'(x)=5*1/x + 3*1/(x-2)`

`color(blue)(f'(x)=5/x +3/(x-2)`

`color(blue)(f'(x)=(5(x-2) +3x)/(x(x-2))`

`color(blue)(f'(x)=(5x-10+3x)/(x(x-2))`

`color(blue)(f'(x)=(8x-10)/(x^2-2x))`