Question

Given that `p(x)=(3/((2x^2-5)))`, find `p''(x)`?

Use differentiation please....

Answer 1

`p''(x) =(72x^2+60)/(2x^2-5)^3`

Explanation:

Hi Loweena.

Here's one way to do this:
I would start by rewriting the equation without the division line:
`p(x)=3*(2x^2-5)^-1`

From that equation, and applying the chain rule, we can find the first derivative:
`p'(x)=-3*(2x^2-5)^-2*4x`
`p'(x)=-12x*(2x^2-5)^-2`

Now we repeat the process, applying the product rule and the chain rule, to get the second derivative:

`p''(x) = -12*(2x^2-5)^-2 + (-12x)(-2)(2x^2-5)^-3*4x`

`p''(x) = -12*(2x^2-5)^-2 + 96x^2*(2x^2-5)^-3`

If you wish, you can simplify this a little bit.
I will factor out `(2x^2-5)^-3`, and simplify what remains:

`p''(x) = (2x^2-5)^-3*[-12*(2x^2-5)^1+96x^2]`

`p''(x) = (2x^2-5)^-3*[-24x^2+60+96x^2]`

`p''(x) =(2x^2-5)^-3*(72x^2+60)`

If you wish, you can also return to the original format:

`p''(x) =(72x^2+60)/(2x^2-5)^3`