Question

What is the derivative of `f(x) = sqrt[ (3 x + 1) / (5 x^2 + 1) ]`?

Answer 1

Use chain rule: if `f(x)=h(g(x))` the derivative is`f'(x)=h'(g(x))*g'(x)`
and quotient rule:
`(f/g)^'={\frac {f'g-fg'}{g^{2}}}` where `g!=0`

Explanation:

`f'(x)=(((3 x + 1)/ (5 x^2 + 1))^(1/2))^'`
`=1/2*((3 x + 1)/ (5 x^2 + 1))^(-1/2)*((3 x + 1)/ (5 x^2 + 1))^'`

Lets calculate `((3 x + 1)/ (5 x^2 + 1))^'` separately

`((3 x + 1)/ (5 x^2 + 1))^'=`
`=((3 x + 1)^'* (5 x^2 + 1)-(3 x + 1)*(5 x^2 + 1)^')/((5 x^2 + 1)^2)=`

`=(3 * (5 x^2 + 1)-(3 x + 1)*10x)/((5 x^2 + 1)^2)`
`=(15x^2+3-30x^2-10x)/((5 x^2 + 1)^2)=`
`=(-15x^2-10x+3)/((5 x^2 + 1)^2)`

The result is:
`f'(x)=(-15x^2-10x+3)/(2(5 x^2 + 1)^2 ((3 x + 1)/ (5 x^2 + 1))^(1/2)`