Question

How do you find the derivative of `(x^(2/3 ))(8-x)`?

Answer 1

`( d (8x^(2/3) - x^(-1/3)))/(dx) = 1/(3root(3)(x)) *(16+1/x)`

Explanation:

`(x^(2/3))(8-x)`
`color(white)("XXXX")` `=8x^(2/3) - x^(-1/3)`

`( d (8x^(2/3) - x^(-1/3)))/(dx) = color(red)((d (8x^(2/3)))/(dx)) - color(blue)((d (x^(-1/3)))/(dx))`

`color(white)("XXXX")` `=color(red)( 2/3*8x^-(1/3)) - color(blue)((-1/3)x^(-4/3))`

`color(white)("XXXX")` `= 16/(3root(3)x) + 1/(3(root(3)(x))^4)`

`color(white)("XXXX")` `= 16/(3root(3)x) + 1/(3(x*root(3)(x)))`

`color(white)("XXXX")` `= 1/(3root(3)(x)) *(16+1/x)`