Question

How to differentiate?? and what rule should be applied here? thanks in advance! :)

Answer 1

You could write this as


`x(x^2-1)^(-1/2)`

Then by the product rule, the derivative is


`(x(x^2-1)^(-1/2))'=x((x^2-1)^(-1/2))'+(x)'(x^2-1)^(-1/2)=x\left(-\frac12(x^2-1)^(-3/2)(2x)\right)+(x^2-1)^(-1/2)`

`\left(\frac x{√(x^2-1)}\right)'=-x^2(x^2-1)^(-3/2)+(x^2-1)^(-1/2)`

Pulling out a factor of
`(x^2-1)^(-3/2)`, the right hand side can be rewritten as


`-(x^2-1)^(-3/2)\left(x^2-(x^2-1)^1\right)=-\frac1{(x^2-1)^(3/2)}`