# How do I find the answer?

Jan 17, 2018

$16 \left(12 {x}^{2} - 1\right)$

#### Explanation:

Differentiate twice! Using chain rule:

$y = {\left(4 {x}^{2} - 1\right)}^{2}$
$\frac{\mathrm{dy}}{\mathrm{dx}} = 2 \cdot \left(4 {x}^{2} - 1\right) \cdot \left(8 x\right) = 64 {x}^{3} - 16 x$

Using power rule,
$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = \frac{d}{\mathrm{dx}} \frac{\mathrm{dy}}{\mathrm{dx}} = 192 {x}^{2} - 16 = 16 \left(12 {x}^{2} - 1\right)$

Jan 17, 2018

Answer 'D' or $\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = 16 \left(12 {x}^{2} - 1\right)$

#### Explanation:

So to find the second order derivative we must differentiate it twice.

I would recommend multiplying out the brackets before doing the differential (it will make the algebra easier in the end)

$y = {\left(4 {x}^{2} - 1\right)}^{2} = 16 {x}^{4} - 8 {x}^{2} + 1$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = 64 {x}^{3} - 16 x$

and $\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = 192 {x}^{2} - 16$

Factorising by 16 yields;

$\frac{{d}^{2} y}{{\mathrm{dx}}^{2}} = 16 \left(12 {x}^{2} - 1\right)$