Question

Differentiating a Logarithmic Function in Exercise, find the derivative of the function. See Examples 1, 2, 3, and 4.

y = ln[(4 + x^2)^1/2]/x

Answer 1

`(dy)/(dx)=(x)/(4+x^(2))`

Explanation:

Given the function:

`y= \ln{(4+x^2)^{(1)/(2)}`

as we know that
`(d)/(dx)(ln((f(x))))=(1)/(f(x))(f'(x))`

`y= \ln{(4+x^2)^{(1)/(2)}`

`(dy)/(dx)= \frac{1}{(4+x^2)^{(1)/(2)}}(d)/(dx)((4+x^2)^{(1)/(2)})`

`(dy)/(dx)=\frac{1}{(4+x^2)^{(1)/(2)}}\left((1)/(2)(4+x^2)^{(1)/(2)-1}(0+2x)\right)`

now we'll just simplify:

`(dy)/(dx)=\frac{1}{(4+x^2)^{(1)/(2)}}\left((1)/(2)(4+x^2)^{-(1)/(2)}(2x)\right)`

`(dy)/(dx)=(x)/((4+x^(2)))`

this is our answer!