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

Answer 1

Aurora Alvarado

The derivative is #=-1/(2(x-1)^(3/2))#

We need

#(x^n)'=nx^(n-1)#

Our function is #f(x)=1/sqrt(x-1)=(x-1)^(-1/2)#

#AA x in (1, +oo)#, #f(x) in (0,+oo)#

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

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Answer 2

Ezekiel Alexander

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

#"express "y=1/(sqrt(x-1))=(x-1)^(-1/2)#

#"differentiate using the "color(blue)"chain rule"#

#"given "y=f(g(x))" then"#

#dy/dx=f'(g(x)xxg'(x)larr" chain rule"#

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

#rArrdy/dx=-1/2(x-1)^(-3/2)xxd/dx(x-1)#

#color(white)(rArrdy/dx)=-1/(2sqrt((x-1)^3))#

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Answer 3

William Abdalla

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

Rewrite the expression as ( (x-1)^{-\frac{1}{2}} ).
Apply the power rule: ( \frac{d}{dx}(x-1)^{-\frac{1}{2}} = -\frac{1}{2}(x-1)^{-\frac{1}{2}-1} ).
Simplify the expression: ( -\frac{1}{2}(x-1)^{-\frac{3}{2}} ).
Therefore, the derivative of ( \frac{1}{\sqrt{x-1}} ) is ( -\frac{1}{2\sqrt{x-1}} ).

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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.