How do you find the derivative of #sqrt(1/x^3)#?

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Answer 1 · 2016-03-15T13:43:04

Rewrite it. #sqrt(1/(x^3)) = x^(-3/2)#. So the derivative is #-3/2x^(-5/2) = (-3)/(2 sqrt(x^5))#

#sqrt(1/(x^3)) = 1/sqrt(x^3) = 1/x^(3/2) = x^(-3/2)#

Now apply the power rule.

If you really like the quotient rule and algebra , then use

#sqrt(1/(x^3)) = 1/sqrt(x^3) = 1/x^(3/2)# and the quotient rule.

So the derivative is #((0)x^(3/2)-1(3/2x^(1/2)))/(x^(3/2))^2#.

Now simplify algebraically to get # (-3)/(2 sqrt(x^5))#.

If you are a glutton for algebra use the chain rule and the quotient rule:

#d/dx(sqrt(1/x^3)) = d/dx((1/x^3)^(1/2))#

# = 1/2 (1/x^3)^(-1/2) [d/dx(1/x^3)]#

# = 1/2 (1/x^3)^(-1/2) [((0)x^3-1(3x^2))/(x^3)^2]#

Now simplify algebraically.