Question

Differentiate 1/√x^2-1

Answer 1

The function is $f(x)= \frac{1}{ \sqrt{ x^{2} -1}}$

write f again as $f(x)= (x^{2} -1)^{- \frac{1}{2} }$ 

(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 $f(x)= (x^{2} -1)^{- \frac{1}{2} }$ we notice 2 functions:

the outer function $u^{ -\frac{1}{2} }$, where $u=x^{2} -1$, and the inner, u itself , which is a function of x.

So we differentiate by using the chain rule:

$f'(x)= -\frac{1}{2}u^{- \frac{1}{2}-1 }*u'= -\frac{1}{2}u^{- \frac{3}{2} }*(2x)=- \frac{x}{ \sqrt{ u^{3} } } =- \frac{x}{ \sqrt{ (x^{2} -1)^{3} } }$


Answer: $- \frac{x}{ \sqrt{ (x^{2} -1)^{3} } }$