How do you find the derivative of

Question

f(s)=sqrt[((s^2) + 1) / ((s^2) + 4)]?

Answer 1

The function f(s) = √[(s² + 1)/(s² + 4)]

Differentiating on each side with respect of s.

Apply chain rule in derivatives.

f'(s) = {1/2√[(s² + 1)/(s² + 4)]} d/ds [(s² + 1)/(s² + 4)]

Apply quotient rule in derivatives d/dx(u/v) = (vu' -uv')/v².

= {1/2√[(s² + 1)/(s² + 4)]} {[(s² + 4)(2s) - (s² + 1)(2s) ]/(s² + 4)²}

= {1/2√[(s² + 1)/(s² + 4)]} [2s³ + 8s - 2s³ - 2s]/(s² + 4)²

f'(s) = {1/2√[(s² + 1)/(s² + 4)]} [6s/(s² + 4)²].