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

Answer 1

It is #-1/(2sqrt((x-1)^3))#.

You can use the chain rule and the derivative of the power.

Your function can be written as

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

we know that the derivative of #x^n# is #nx^(n-1)#. In this case we do not have #x^(-1/2)# but we have #(x-1)^(-1/2)#. So we have to apply the chain rule and write

#d/dx(x-1)^(-1/2)=-1/2(x-1)^(-1/2-1)*d/dx(x-1)#

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

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

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Answer 2

Christopher Agresta

To find the derivative of ( \frac{1}{\sqrt{x-1}} ), you would use the power rule and the chain rule.

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

Therefore, the derivative of ( \frac{1}{\sqrt{x-1}} ) is ( -\frac{1}{2\sqrt{(x-1)^3}} ).

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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.