# How do you find the derivative of 1 / (2*sqrt(x+1))?

Jul 1, 2017

Unsimplified, it's
$- {\left(2 \cdot \setminus \sqrt{x + 1}\right)}^{-} 2 \cdot 2 \cdot \frac{1}{2} \cdot {\left(x + 1\right)}^{- \frac{1}{2}}$
Simplified, it's
$- \setminus \frac{1}{\left(4 x + 4\right) \left(\setminus \sqrt{x + 1}\right)}$
Creatively simplified, it's
$- \setminus \frac{1}{4 {\left(\setminus \sqrt{x + 1}\right)}^{3}}$

#### Explanation:

You can do this in two ways. You can either use the quotient rule and then chain rule to differentiate, or you can simplify the expression and use the power rule and chain rule to differentiate. I'm going to use the power rule, since it's a bit less messy:

$\frac{d}{\mathrm{dx}} \left[\setminus \frac{1}{2 \cdot \setminus \sqrt{x + 1}}\right]$
$= \frac{d}{\mathrm{dx}} \left[{\left(2 \cdot \setminus \sqrt{x + 1}\right)}^{-} 1\right]$
$= - {\left(2 \cdot \setminus \sqrt{x + 1}\right)}^{-} 2 \cdot \frac{d}{\mathrm{dx}} \left[2 \cdot \setminus \sqrt{x + 1}\right]$
$= - {\left(2 \cdot \setminus \sqrt{x + 1}\right)}^{-} 2 \cdot 2 \cdot \frac{d}{\mathrm{dx}} \left[{\left(x + 1\right)}^{\frac{1}{2}}\right]$
$= - {\left(2 \cdot \setminus \sqrt{x + 1}\right)}^{-} 2 \cdot 2 \cdot \frac{1}{2} \cdot {\left(x + 1\right)}^{- \frac{1}{2}}$

The above would be the answer, but simplifying this expression results in:
$= - \setminus \frac{1}{{\left(2 \cdot \setminus \sqrt{x + 1}\right)}^{2}} \cdot \setminus \frac{1}{\setminus \sqrt{x + 1}}$
$= - \setminus \frac{1}{4 \left(x + 1\right)} \setminus \cdot \setminus \frac{1}{\setminus \sqrt{x + 1}}$
$= - \setminus \frac{1}{\left(4 x + 4\right) \left(\setminus \sqrt{x + 1}\right)}$

Or, if you want to get creative with it,
$= - \setminus \frac{1}{4 {\left(\setminus \sqrt{x + 1}\right)}^{3}}$

Jul 3, 2017

Yes I get the same answer $- \frac{1}{4} {\left[x + 1\right]}^{- \frac{3}{2}}$

#### Explanation:

But I would convert to fractional power notation immediately.
$\left(\frac{1}{2}\right) {\left(x + 1\right)}^{- \frac{1}{2}}$
The the derivative is found with the Power Rule
$\left(- \frac{1}{2}\right) \left(\frac{1}{2}\right) {\left(x + 1\right)}^{- \frac{3}{2}}$
$\left(- \frac{1}{4}\right) \left(\frac{1}{{\left(x + 1\right)}^{3}} ^ \left(\frac{1}{2}\right)\right)$

I find Root symbols cumbersome to work with compared to fractional power notation.