Question

What is the derivative of `(x^(2/3 ))(8-x)`?

Answer 1

`[(x^(2/3))(8-x)]'=(16-5x)/(3root(3)(x))`

Explanation:

Recall the following:

`1`. Product Rule: `[f(x)g(x)]'=f(x)g'(x)+f'(x)g(x)`
`2`. Difference Rule: `[f(x)-g(x)]'=f'(x)-g'(x)`
`3`. Constant Rule: `c'=0`
`4`. Power Rule: `(x^n)'=nx^(n-1)`

Finding the Derivative
`1`. Start by rewriting the equation using the product rule.

`[(x^(2/3))(8-x)]'`

`=(x^(2/3))(color(red)(8-x))'+(color(purple)(x^(2/3)))'(8-x)`

`2`. Use the difference rule and constant rule for `(color(red)(8-x))`. Recall that the derivative of a constant is always `color(orange)0` and the derivative of `x` is always `color(blue)1`.

`=(x^(2/3))(color(red)(8'-x'))+(color(purple)(x^(2/3)))'(8-x)`

`=(x^(2/3))(color(orange)0-color(blue)1)+(color(purple)(x^(2/3)))'(8-x)`

`3`. Use the power rule for `color(purple)(x^(2/3))`.

`=(x^(2/3))(0-1)+(2/3x^(-1/3))(8-x)`

`4`. Simplify.

`=-x^(2/3)+(2/(3x^(1/3)))(8-x)`

`=(2/(3root(3)(x)))(8-x)-x^(2/3)`

`=(16-2x)/(3root(3)(x))-x^(2/3)`

`=(16-2x)/(3root(3)(x))-(x^(2/3)(3root(3)(x)))/(3root(3)(x))`

`=(16-2x-x^(2/3)(3root(3)(x)))/(3root(3)(x))`

`=(16-2x-x^(2/3)(3x^(1/3)))/(3root(3)(x))`

`=(16-2x-3x)/(3root(3)(x))`

`=color(green)((16-5x)/(3root(3)(x)))`

`:.`, the derivative is `(16-5x)/(3root(3)(x))`.