Question

Can someone pls help me find the second derivative of this. I need to find the concave up and down intervals but I’m having trouble getting the second derivative

Answer 1

Starting from

`f(x)=\frac14(x+1)^(\frac83)-4(x+1)^(\frac23)`

take the first derivative using the power and chain rules:

`f'(x)=\frac83\cdot\frac14(x+1)^(\frac83-1)-\frac23\cdot4(x+1)^(\frac23-1)`

`f'(x)=\frac23(x+1)^(\frac53)-\frac83(x+1)^(-\frac13)`

Now take the second derivative:

`f''(x)=\frac53\cdot\frac23(x+1)^(\frac53-1)-\left(-\frac13\right)\cdot\frac83(x+1)^(-\frac13-1)`

`f''(x)=\boxed{\frac{10}9(x+1)^(\frac23)+\frac89(x+1)^(-\frac43)}`

Optionally, you can condense the second derivative a bit by factoring out
`\frac89(x+1)^(-\frac43)`, which gives

`f''(x)=\frac89 (x+1)^(-\frac43) \left(80(x+1)^(\frac63)+1\right)`

`f''(x)=\frac89 (x+1)^(-\frac43) \left(80(x+1)^2+1\right)`

`f''(x)=\frac89 (x+1)^(-\frac43) \left(80x^2+160x+81\right)`