Question

What is the derivative of `1/sqrt(1 - x^2)`?

Answer 1

`=(x)(1-x^2)^(-3/2)`

Explanation:

`f(x)=1/sqrt(1-x^2)=(sqrt(1-x^2))^-1=(1-x^2)^(-1/2)`
`d/dx f(x)=-1/2(1-x^2)^(-1/2-1)(0-2x)`
`=(2x)/2(1-x^2)^(-3/2)`
`=(x)(1-x^2)^(-3/2)`

Answer 2

The derivative is `(xsqrt(1-x^2))/(1-x^2)^2`.

Explanation:

Using the quotient rule:

`color(white)=d/dx[1/sqrt(1-x^2)]`

`=(d/dx[1]*sqrt(1-x^2)-1*d/dx[sqrt(1-x^2)])/(sqrt(1-x^2))^2`

`=(0*sqrt(1-x^2)-1*d/dx[sqrt(1-x^2)])/(1-x^2)`

`=(-d/dx[sqrt(1-x^2)])/(1-x^2)`

Chain rule:

`=(-1/(2sqrt(1-x^2))*d/dx[1-x^2])/(1-x^2)`

`=(-1/(2sqrt(1-x^2))*-2x)/(1-x^2)`

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

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

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

That's the derivative. Hope this helped!