Possible derivation:
d/dx(sqrt(x/(1 + x^2)))
Using the chain rule, d/dx(sqrt(x/(x^2 + 1))) = ( dsqrt(u))/( du) 0, where u = x/(x^2 + 1) and ( d)/( du)(sqrt(u)) = 1/(2 sqrt(u)):
= (d/dx(x/(1 + x^2)))/(2 sqrt(x/(1 + x^2)))
Use the quotient rule, d/dx(u/v) = (v ( du)/( dx) - u ( dv)/( dx))/v^2, where u = x and v = x^2 + 1:
= ((x^2 + 1) d/dx(x) - x d/dx(1 + x^2))/((x^2 + 1)^2) 1/(2 sqrt(x/(1 + x^2)))
The derivative of x is 1:
= (-x (d/dx(1 + x^2)) + 1 (1 + x^2))/(2 sqrt(x/(1 + x^2)) (1 + x^2)^2)
Simplify the expression:
= (1 + x^2 - x (d/dx(1 + x^2)))/(2 sqrt(x/(1 + x^2)) (1 + x^2)^2)
Differentiate the sum term by term:
= (1 + x^2 - d/dx(1) + d/dx(x^2) x)/(2 sqrt(x/(1 + x^2)) (1 + x^2)^2)
The derivative of 1 is zero:
= (1 + x^2 - x (d/dx(x^2) + 0))/(2 sqrt(x/(1 + x^2)) (1 + x^2)^2)
Simplify the expression:
= (1 + x^2 - x (d/dx(x^2)))/(2 sqrt(x/(1 + x^2)) (1 + x^2)^2)
Use the power rule, d/dx(x^n) = n x^(n - 1), where n = 2: d/dx(x^2) = 2 x:
= (1 + x^2 - 2 x x)/(2 sqrt(x/(1 + x^2)) (1 + x^2)^2)
Simplify the expression:
Answer: |= (1 - x^2)/(2 sqrt(x/(1 + x^2)) (1 + x^2)^2)
