# Examine if function f(x)=ln(x+sq root(1+x^2)) increasing?

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### 1 Answer

You should use the first derivative of the function to test if the function increases all over its domain.

`f'(x) = (ln(x + sqrt(1 + x^2)))'`

You need to evaluate the derivative of the function using the chain rule, such that:

`f'(x) =1/(x + sqrt(1 + x^2))*(x + sqrt(1 + x^2))'`

`f'(x) =1/(x + sqrt(1 + x^2))*(1 + (2x)/(2sqrt(1 + x^2)))`

`f'(x) =1/(x + sqrt(1 + x^2))*(1 + x/(sqrt(1 + x^2)))`

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

You should solve for x the equation `f'(x) = 0` , such that:

`f'(x) = 0 => sqrt(1 + x^2) + x = 0 => sqrt(1 + x^2) = -x` invalid, hence `f'(x) > 0` for `x in R` .

**Hence, since `f'(x) > 0` for all `x in R,` yields that `f(x) ` increases over the set `R` .**

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