# How do you differentiate  y =sqrt((3x-9)^3  using the chain rule?

##### 1 Answer
Jan 8, 2016

$h ' \left(x\right) = \frac{d}{\mathrm{dx}} \sqrt{{\left(3 x - 9\right)}^{3}} = \frac{9}{2} \sqrt{3 x - 9}$

#### Explanation:

given

$h \left(x\right) = f \left(g \left(x\right)\right) = \left(f \circ g\right)$

The chain rule is

$h ' \left(x\right) = \left(f \circ g\right) ' = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

$h \left(x\right) = y = \sqrt{{\left(3 x - 9\right)}^{3}} = {\left({\left(3 x - 9\right)}^{3}\right)}^{\frac{1}{2}} = {\left(3 x - 9\right)}^{\frac{3}{2}}$

then
$f \left(x\right) = {\left(\right)}^{\frac{3}{2}}$

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

$g \left(x\right) = \left(3 x - 9\right)$

$g ' \left(x\right) = 3$

Apllying the rule

$h ' \left(x\right) = \frac{3}{2} {\left(3 x - 9\right)}^{\frac{3}{2} - 1} \cdot 3 = \frac{9}{2} {\left(3 x - 9\right)}^{\frac{3 - 2}{2}} = \frac{9}{2} {\left(3 x - 9\right)}^{\frac{1}{2}} =$
$\frac{9}{2} \sqrt{3 x - 9}$

Alternatively:

$f \left(x\right) = \sqrt{}$

$f ' \left(x\right) = \frac{1}{2 \cdot \sqrt{}}$

$g \left(x\right) = {\left(3 x - 9\right)}^{3}$

it is a composite function too

$g ' \left(x\right) = 3 {\left(3 x - 9\right)}^{3 - 1} \cdot \frac{d}{\mathrm{dx}} \left(3 x - 9\right) = 3 {\left(3 x - 9\right)}^{2} \cdot 3 =$
$= 9 {\left(3 x - 9\right)}^{2}$

then:

$h ' \left(x\right) = \left(\frac{1}{2 \cdot \sqrt{{\left(3 x - 9\right)}^{3}}}\right) 9 \cdot {\left(3 x - 9\right)}^{2} =$
$= \frac{9}{2} \cdot \left({\left(3 x - 9\right)}^{2} / \left(\sqrt{{\left(3 x - 9\right)}^{2} \cdot \left(3 x - 9\right)}\right)\right) =$
$= \frac{9}{2} \cdot \left({\left(3 x - 9\right)}^{\textcolor{b l u e}{\cancel{2}}} / \left(\textcolor{b l u e}{\cancel{\left(3 x - 9\right)}} \cdot \sqrt{3 x - 9}\right)\right) =$
$= \frac{9}{2} \frac{3 x - 9}{\sqrt{3 x - 9}} = \frac{9}{2} \sqrt{{\left(3 x - 9\right)}^{\textcolor{b l u e}{\cancel{2}}} / \textcolor{b l u e}{\cancel{\left(3 x - 9\right)}}} =$
$= \frac{9}{2} \sqrt{3 x - 9}$