the function is [tex]f(x)= \frac{1}{ \sqrt{ x^{2} -1}} [/tex]
write f again as [tex]f(x)= (x^{2} -1)^{- \frac{1}{2} } [/tex]
(the power -1 takes the expression to the denominator, and the power 1/2 is square root)
writing rational expressions as power expressions, generally makes differentiation more practical.
In [tex]f(x)= (x^{2} -1)^{- \frac{1}{2} } [/tex] we notice 2 functions:
the outer function [tex]u^{ -\frac{1}{2} } [/tex], where [tex]u=x^{2} -1[/tex], and the inner, u itself , which is a function of x.
So we differentiate by using the chain rule:
[tex]f'(x)= -\frac{1}{2}u^{- \frac{1}{2}-1 }*u'= -\frac{1}{2}u^{- \frac{3}{2} }*(2x)=- \frac{x}{ \sqrt{ u^{3} } } =- \frac{x}{ \sqrt{ (x^{2} -1)^{3} } }[/tex]
Answer: [tex]- \frac{x}{ \sqrt{ (x^{2} -1)^{3} } }[/tex]
