# How do you differentiate sqrt(4x² + 1)?

Jun 5, 2015

We'll use the power rule and the chain rule.

$f \left(x\right) = \sqrt{4 {x}^{2} + 1} = {\left(4 {x}^{2} + 1\right)}^{\frac{1}{2}}$

So,

$f ' \left(x\right) = \frac{1}{2} {\left(4 {x}^{2} + 1\right)}^{\frac{1}{2} - 1} \cdot \frac{d}{\mathrm{dx}} \left(4 {x}^{2} + 1\right)$

$= \frac{1}{2} {\left(4 {x}^{2} + 1\right)}^{- \frac{1}{2}} \cdot 8 x$

$= \frac{1}{2 {\left(4 {x}^{2} + 1\right)}^{\frac{1}{2}}} \cdot \frac{8 x}{1}$

$= \frac{4 x}{\sqrt{4 {x}^{2} + 1}}$

And the last line cannot be simplified any further, so we're finished.