Question

Differentiate the following with respect to x.

`1/sqrt(1 + x^2)`

Answer 1

For the following problems chain rule to be used:

`"d"/"dx"` f(g(x)) = f'(g(x)) . g'(x)

`"d"/"dx"` [f(x)]ⁿ = n[f(x)]^n-1 × `"d"/"dx"`f(x)

Let y = `1/sqrt(1 + x^2)` 

y = `(1 + x^2)^(- 1/2)`

Here n = `- 1/2`; f(x) = 1 + x²

`"dy"/"dx" = - 1/2(1 + x^2)^(- 1/2 - 1) "d"/"dx" (1 + x^2)`

`= - 1/2 (1 + x^2)^(- 3/2)`(0 + 2x)

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

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

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

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