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

Answer 1

Luke Alvarez

First write the function as #x^-(3/2)#

As we know# (x^a")'=ax^(a-1)#:

Then #x^-(3/2)'=-3/2x^(-5/2)#

or #-3/(2sqrt(x^5))#

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Answer 2

Aurora Alvarado

To find the derivative of ( \sqrt{\frac{1}{x^3}} ), you can use the chain rule.

Start by expressing the function as ( \left(\frac{1}{x^3}\right)^{1/2} ). Then, differentiate the function with respect to ( x ) using the chain rule:

[ \frac{d}{dx}\left(\sqrt{\frac{1}{x^3}}\right) = \frac{1}{2} \left(\frac{1}{x^3}\right)^{-1/2} \cdot \left(-3x^{-4}\right) ]

Simplify this expression:

[ = \frac{1}{2} \cdot \frac{1}{\sqrt{x^3}} \cdot (-3x^{-4}) = -\frac{3}{2} \cdot \frac{1}{x^2\sqrt{x}} ]

So, the derivative of ( \sqrt{\frac{1}{x^3}} ) with respect to ( x ) is ( -\frac{3}{2x^2\sqrt{x}} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.