Question

How do you differentiate `f(x) = ln [ x^{2}(x + 3)^{8}(x^2 + 4)^{3} ]`?

Answer 1

I'll leave it to someone else to use `d/dx(lnu) = 1/u (du)/dx` and the three factor product rule. I'll do it the easy way:

`f(x) = ln [ x^{2}(x + 3)^{8}(x^2 + 4)^{3} ]`

`f(x) = ln x^2+ln(x + 3)^8 + ln(x^2 + 4)^3`

`f(x) =2 ln x + 8 ln(x + 3) +3 ln(x^2 + 4)`

So
`f'(x) = 2/x + 8/(x+3) +3/(x^2+4) *(2x)`

`f'(x) = 2/x + 8/(x+3) +(6x)/(x^2+4)`.