# What is the derivative of the following function? Thank you!

Jun 24, 2018

$18 {x}^{2} - 9 x + 9 = 9 \left(2 {x}^{2} - x + 1\right)$

#### Explanation:

Assuming you have: $f \left(x\right) = 3 x \left(4 {x}^{2} - 3 x + 6\right) \frac{1}{2}$.

We first simplify the expression into:

$f \left(x\right) = 1.5 x \left(4 {x}^{2} - 3 x + 6\right)$

Now, we use the product rule, which states that:

$\frac{d}{\mathrm{dx}} \left(u v\right) = u ' v + u v '$

Let $u = 1.5 x , \therefore u ' = 1.5$.

Then $v = 4 {x}^{2} - 3 x + 6 , v ' = 8 x - 3$ by the power rule.

Substituting back into the product rule formula, we get:

$f ' \left(x\right) = 1.5 \left(4 {x}^{2} - 3 x + 6\right) + 1.5 x \left(8 x - 3\right)$

$= 6 {x}^{2} - 4.5 x + 9 + 12 {x}^{2} - 4.5 x$

$= 18 {x}^{2} - 9 x + 9$