Question

What is the first derivative and second derivative of `4x^(1/3)+2x^(4/3)`?

Answer 1

`(d y)/(d x)=4/3*x^(-2/3)+8/3*x^(1/3)" (the first derivative)"`
`(d^2 y)/(d t^2)=8/9*x^(-2/3)(-x^-1+1)" (the second derivative)"`

Explanation:

`y=4x^(1/3)+2x^(4/3)`

`(d y)/(d x)=1/3*4*x^((1/3-1))+4/3*2x^((4/3-1))`

`(d y)/(d x)=4/3*x^(-2/3)+8/3*x^(1/3)" (the first derivative)"`

`(d^2 y)/(d t^2)=-2/3*4/3*x^((-2/3-1))+8/3*1/3*x^((1/3-1))`

`(d^2 y)/(d t^2)=-8/9*x^((-5/3))+8/9*x^((-2/3)`

`(d^2 y)/(d t^2)=8/9*x^(-2/3)(-x^-1+1)" (the second derivative)"`