How do you find the derivative of #sqrt(x^2-1)#?

1 Answer
Ken C.
Mar 13, 2016

Use the power rule and chain rule to get #x/(sqrt(x^2-1))#.

Explanation:

Recognize that #sqrt(x^2-1)# can be written as #(x^2-1)^(1/2)#. Now that we have it in this form, we can use the power rule to get:
#d/dx=1/2(x^2-1)^(-1/2)#

We also need to use the chain rule; the derivative of the "inside" function (#x^2#) is #2x#, so we multiply #1/2(x^2-1)^(-1/2)# by #2x#:
#d/dx=2x*1/2(x^2-1)^(-1/2)#

Canceling the #2#s and using the properties of exponents yields:
#d/dx=x/((x^2-1)^(1/2))#

Since the problem was given to us in radical form, convert back to get the final answer:
#d/dx=x/(sqrt(x^2-1))#

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