Question

How to differentiate this equation?

Answer 1

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

Explanation:

.

`f(x)=log_esqrt(x^2+1)`

A `log` with base `e` is a natural log. Therefore,

`f(x)=lnsqrt(x^2+1)`

Let `u=x^2+1`

`du=2xdx, :. (du)/dx=2x`

`f(u)=lnsqrtu`

Let `z=sqrtu`

`dz=1/2u^(-1/2)du=1/(2u^(1/2))du=1/(2sqrtu)du=sqrtu/(2u)du`

`(dz)/(du)=sqrtu/(2u)`

`f(z)=lnz`

`dy/(dz)=1/z`

The Chain Rule says:

`dy/dx=(dy/(dz))((dz)/(du))((du)/dx)`

`dy/dx=(1/z)(sqrtu/(2u))(2x)=(1/sqrtu)(sqrtu/(2u))(2x)=x/u`

Now, we can substitute back for `u` and `z`:

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