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

$f(x)=\dfrac14(x+1)^{\frac83}-4(x+1)^{\frac23}$

take the first derivative using the power and chain rules:

$f'(x)=\dfrac83\cdot\dfrac14(x+1)^{\frac83-1}-\dfrac23\cdot4(x+1)^{\frac23-1}$

$f'(x)=\dfrac23(x+1)^{\frac53}-\dfrac83(x+1)^{-\frac13}$

Now take the second derivative:

$f''(x)=\dfrac53\cdot\dfrac23(x+1)^{\frac53-1}-\left(-\dfrac13\right)\cdot\dfrac83(x+1)^{-\frac13-1}$

$f''(x)=\boxed{\dfrac{10}9(x+1)^{\frac23}+\dfrac89(x+1)^{-\frac43}}$

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

$f''(x)=\dfrac89 (x+1)^{-\frac43} \left(80(x+1)^{\frac63}+1\right)$

$f''(x)=\dfrac89 (x+1)^{-\frac43} \left(80(x+1)^2+1\right)$

$f''(x)=\dfrac89 (x+1)^{-\frac43} \left(80x^2+160x+81\right)$