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