Question

What is the second derivative of? : `2(x^2-1)^3`

Answer 1

`12(x^2-1)^2 + 48x^2(x^2-1)`

Explanation:

To find the second derivative, you need to find the first derivative, so let's do that immediately:

`d/dx 2(x^2-1)^3 = 6(x^2-1)^2 * (2x) = 12x(x^2-1)^2`

I used the chain rule and product rule to find the first derivative. Now let's find the second derivative:

#d/dx 12x(x^2-1)^2 = 12(x^2-1)^2+12x*[2(x^2-1)*2x] = 12(x^2-1)^2 + 48x^2(x^2-1)#

For the second derivative, I used the product rule and chain rule.

Answer 2

`(d^2)/(dx^2) 2(x^2-1)^3 = 12(x^2-1)(5x^2-1)`

Explanation:

First we generate the first derivative using the chain rule:

`d/dx 2(x^2-1)^3 = 2(3)2(x^2-1)^2 d/dx ( x^2-1)`

`" " = 6(x^2-1)^2 (2x)`

`" " = 12x(x^2-1)^2`

Then, we generate the second derivative using the product rule and the chain rule:

`(d^2)/(dx^2) 2(x^2-1)^3 = d/dx 12x(x^2-1)^2`
`" " = 12x(d/dx (x^2-1)^2) + (d/dx12x)(x^2-1)^2`
`" " = 12x(2(x^2-1)(d/dx(x^2-1))) + 12(x^2-1)^2`
`" " = 24x(x^2-1)(2x) + 12(x^2-1)^2`

`" " = 12(x^2-1){4x^2 + (x^2-1)}`
`" " = 12(x^2-1)(5x^2-1)`