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

Aug 16, 2015

This can be re-written to make it simpler. I hope you understand my method :)

#### Explanation:

You can re-write it as this:

$\sqrt{{x}^{2} + 1}$ = ${\left({x}^{2} + 1\right)}^{\frac{1}{2}}$

After simplifying it, it is easy to apply the derivative rules.

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

$=$ $\frac{1}{2}$${\left({x}^{2} + 1\right)}^{- \frac{1}{2}}$ $\cdot$ $2 x$

$=$ $\frac{1}{2} \cdot$ $\frac{1}{\sqrt{{x}^{2} + 1}}$ $\cdot$ $2 x$

$=$ $\frac{2 x}{2 \sqrt{{x}^{2} + 1}}$

$=$ $\frac{x}{\sqrt{{x}^{2} + 1}}$