Question

How do you differentiate `ln(x^2+1)^(1/2)`?

Answer 1

`frac{"d"}{"d"x}(ln(x^2+1)^{1/2}) = frac{x}{x^2 + 1} * ln(x^2+1)^{-1/2}`

Explanation:

You need to use the chain rule twice. We do this from inside out.

First let `w = x^2 + 1`.

We differentiate `w` w.r.t. `x` using the power rule.

`frac{"d"w}{"d"x} = 2x`

Next, let `v = ln(x^2+1) = ln(w)`

To differentiate `v` w.r.t. `x`, we use the chain rule.

`frac{"d"v}{"d"x} = frac{"d"v}{"d"w} * frac{"d"w}{"d"x}`

`= frac{"d"}{"d"w}(ln(w)) * (2x)`

`= 1/w * (2x)`

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

Finally, we let `u = ln(x^2+1)^{1/2} = sqrt(v)`. `quad frac{"d"u}{"d"x}` is the derivative that we are seeking. We use the chain rule again.

`frac{"d"u}{"d"x} = frac{"d"u}{"d"v} * frac{"d"v}{"d"x}`

`= frac{"d"}{"d"v}(sqrt(v)) * (2x)/(x^2 + 1)`

`= frac{1}{2sqrt(v)} * (2x)/(x^2 + 1)`

`= frac{1}{2sqrt(ln(x^2+1))} * (2x)/(x^2 + 1)`

`= frac{x}{x^2 + 1} * ln(x^2+1)^{-1/2}`