How do you differentiate f(x)=5(3x^2+1)^(1/2) (3x+1) using the product rule?

Nov 9, 2015

Product rule:
if $f \left(x\right) = g \left(x\right) \cdot h \left(x\right)$, then $f ' \left(x\right) = g ' \left(x\right) \cdot h \left(x\right) + h ' \left(x\right) \cdot g \left(x\right)$.

Here, $g \left(x\right) = 5 {\left(3 {x}^{2} + 1\right)}^{\frac{1}{2}}$ and $h \left(x\right) = 3 x + 1$.

Now you need to differentiate $g \left(x\right)$ and $h \left(x\right)$.

$h \left(x\right) = 3 x + 1$
$h ' \left(x\right) = 3$

$g \left(x\right)$ is the complicated one because there, you will need the chain rule.
$g \left(x\right) = 5 {\left(3 {x}^{2} + 1\right)}^{\frac{1}{2}} = 5 u {\left(x\right)}^{\frac{1}{2}}$ where $u \left(x\right) = 3 {x}^{2} + 1$.

With the chain rule,
$g ' \left(x\right) = 5 \cdot \left(\frac{1}{2}\right) \cdot u {\left(x\right)}^{- \frac{1}{2}} \cdot u ' \left(x\right)$
$= \frac{5}{2} \cdot {\left(3 {x}^{2} + 1\right)}^{- \frac{1}{2}} \cdot 6 x$
$= 15 x \cdot {\left(3 {x}^{2} + 1\right)}^{- \frac{1}{2}}$
$= \frac{15 x}{3 {x}^{2} + 1} ^ \left(\frac{1}{2}\right)$

So, we have
$g \left(x\right) = 5 {\left(3 {x}^{2} + 1\right)}^{\frac{1}{2}}$
$g ' \left(x\right) = 15 x \cdot {\left(3 {x}^{2} + 1\right)}^{- \frac{1}{2}}$

Last thing left to do is to apply the product rule:

$f ' \left(x\right) = g ' \left(x\right) \cdot h \left(x\right) + h ' \left(x\right) \cdot g \left(x\right)$
$= 15 x \cdot {\left(3 {x}^{2} + 1\right)}^{- \frac{1}{2}} \cdot \left(3 x + 1\right) + 3 \cdot 5 {\left(3 {x}^{2} + 1\right)}^{\frac{1}{2}}$
$= \frac{15 x \left(3 x + 1\right)}{\sqrt{3 {x}^{2} + 1}} + 15 \sqrt{3 {x}^{2} + 1}$
$= \frac{15 x \left(3 x + 1\right)}{\sqrt{3 {x}^{2} + 1}} + \frac{15 \sqrt{3 {x}^{2} + 1} \cdot \sqrt{3 {x}^{2} + 1}}{\sqrt{3 {x}^{2} + 1}}$
$= \frac{45 {x}^{2} + 15 x + 15 \left(3 {x}^{2} + 1\right)}{\sqrt{3 {x}^{2} + 1}}$
$= \left(90 {x}^{2} + 15 x + 15\right) \cdot {\left(3 {x}^{2} + 1\right)}^{\frac{1}{2}}$
$= 15 \left(6 {x}^{2} + x + 1\right) \cdot {\left(3 {x}^{2} + 1\right)}^{\frac{1}{2}}$

I hope that this helped!