What is the derivative of sqrt(2x) ?

Jan 29, 2016

$\frac{1}{\sqrt{2 x}}$

Explanation:

The function can be rewritten as

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

To differentiate this, use the power rule and chain rule.

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

Differentiating with the power rule gives the $\frac{1}{2} {\left(2 x\right)}^{- \frac{1}{2}}$ part, and through the chain rule you must multiply this by the derivative of the internal function, which is $2 x$.

This gives:

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

The $2$s will cancel.

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