Question

How do you differentiate `log_3(9x^2sin(9x^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`

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

`y=log_3(9x^2sin(9x^2))`

`y=log_3 9 + log_3x^2+log_3 sin(9x^2)`

`y=log_3 9 + 2 log_3x+log_3 sin(9x^2)`

`color(blue)(y'=0+2/(xln3)+1/(sin(9x^2)ln3) * cos (9x^2)*18x`

`color(blue)(y'=2/(xln3)+(cos (9x^2)*18x)/(sin(9x^2)ln3)`

`color(blue)(y'=2/(xln3)+((18x) cot(9x^2))/ln3`

`color(blue)(y'=((18x^2) cot(9x^2))/(xln3)`