Question

Differentiate the following with respect to x.

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

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

Here f(x) = 1 + x²; n = `1/2`

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

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

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

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

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