# Find the derivative of this function?

## $y = x \left({x}^{2} - 3\right)$

Jan 10, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = 3 \left({x}^{2} - 1\right)$

#### Explanation:

Since this is problem in the Product Rule section we will apply it here;

The Product Rule states that:

If $y = f \left(x\right) \cdot g \left(x\right)$
Then $\frac{\mathrm{dy}}{\mathrm{dx}} = f \left(x\right) \cdot g ' \left(x\right) + f ' \left(x\right) \cdot g \left(x\right)$

In our example:

$f \left(x\right) = x$ and $g \left(x\right) = \left({x}^{2} - 3\right)$

Hence: $\frac{\mathrm{dy}}{\mathrm{dx}} = x \cdot 2 x + 1 \cdot \left({x}^{2} - 3\right)$

$= 2 {x}^{2} + {x}^{2} - 3$

$= 3 \left({x}^{2} - 1\right)$

Note however that this problem is more simply solved by first expanding the expression and applying the Power Rule as follows:

$y = x \left({x}^{2} - 3\right)$

$= {x}^{3} - 3 x$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 3 {x}^{2} - 3$

$= 3 \left({x}^{2} - 1\right)$