Question

How do you find the derivative of `x^(5/3) * ln(3x)`?

Answer 1

`x^(2/3)(1+5/3ln(3x))`

Explanation:

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

If f(x) = g(x).h(x) then f'(x) = g(x)h'(x) + h(x)g'(x)...(A)
`"-----------------------------------------------------"`

`g(x)=x^(5/3)rArrg'(x)=5/3x^(2/3)-"using the"color(blue)" power rule"`
`h(x)=ln(3x)rArrh'(x)=1/(3x)xx3=1/x`

using `d/dx(ln(f(x)))=1/(f(x))xxd/dxf(x)`
`"---------------------------------------------------------"`
Substitute these values into (A)

`f'(x)=x^(5/3)xx1/x+ln(3x)xx5/3x^(2/3)`

`=x^(5/3-1)+5/3x^(2/3)ln(3x)=x^(2/3)+5/3x^(2/3)ln(3x)`

Taking out a common factor of `x^(2/3)`

`rArrd/dx(x^(5/3).ln(3x))=x^(2/3)(1+5/3ln(3x))`