Question

Could you differentiate this function? f(x)=6x^2ln(4x)

Answer 1

`12x*ln(4x)+6x`

Explanation:

first use product rule:

`f'(x) = ln(4x)*d/dx(6x^2)+6x^2*d/dx(ln(4x))`

`=ln(4x)*12x+(6x^2)/(4x)*d/dx(4x)` (power rule, ln(x) derivative, and chain rule)

`=ln(4x)*12x+(3/2x*4)`
`=12x*ln(4x)+6x`

Answer 2

`f'(x)=12(1+ln4x)`

Explanation:

`"differentiate using the "color(blue)"product rule"`

`"given "f(x)=g(x)h(x)" then"`

`f'(x)=g(x)h'(x)+h(x)g'(x)larrcolor(blue)"product rule"`

`g(x)=6x^2rArrg'(x)=12x`

.`h(x)=ln4xrArrh'(x)=1/(4x)xxd/dx(4x)=1/(4x) xx4=1/x`

`rArrf'(x)=6x^2 xx1/x+ln4x.12x`

`=6x+12xln4x`

`=6x(1+2ln4x)`