How do you find the derivative of 1/sqrt(x)?

Jun 5, 2016

This function can be written as a composition of two functions, therefore we use the chain rule.

Explanation:

Let $f \left(x\right) = \frac{1}{\sqrt{x}}$, then $y = \frac{1}{u} \mathmr{and} u = {x}^{\frac{1}{2}}$, since $\sqrt{x} = {x}^{\frac{1}{2}}$.

Simplifying further, we have that $y = u \mathmr{and} u = {x}^{- \frac{1}{2}}$

The chain rule states $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \times \frac{\mathrm{du}}{\mathrm{dx}}$

This means we have to differentiate both functions and multiply them. Let's start with $y$.

By the power rule $y ' = 1 \times {u}^{0} = 1$.

Now for $u$:

Once again by the power rule we get:

$u ' = - \frac{1}{2} \times {x}^{- \frac{1}{2} - 1}$

$u ' = - \frac{1}{2} {x}^{- \frac{3}{2}}$

$u ' = - \frac{1}{2 \sqrt{{x}^{3}}}$

$f ' \left(x\right) = y ' \times u '$

$f ' \left(x\right) = 1 \times - \frac{1}{2 \sqrt{{x}^{3}}}$

$f ' \left(x\right) = - \frac{1}{2 \sqrt{{x}^{3}}}$

Hopefully this helps!