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

Question

283688 views around the world

You can reuse this answer
Creative Commons License

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!