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

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Answer 1 · 2016-06-05T14:50:56

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

Let $f(x) = 1/sqrt(x)$ , then $y = 1/u and u = x^(1/2)$ , since $sqrt(x) = x^(1/2)$ .

Simplifying further, we have that $y = u and u = x^(-1/2)$

The chain rule states $dy/dx = dy/(du) xx (du)/dx$

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

By the power rule $y' = 1 xx u^0 = 1$ .

Now for $u$ :

Once again by the power rule we get:

$u' = -1/2 xx x^(-1/2 - 1)$

$u' = -1/2x^(-3/2)$

$u' = -1/(2sqrt(x^3))$

$f'(x) = y' xx u'$

$f'(x) = 1 xx -1/(2sqrt(x^3))$

$f'(x) = -1/(2sqrt(x^3))$

Hopefully this helps!

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