What is the derivative of x over 1 plus x squared?

Answer 1

The product rule states: d/dx uv = vdu/dx + udv/dx

It is not quite clear what your denominator is:

• "over 1 + (x squared)": d/dx (x/1+x2)

d/dx(x/1+x2) = d/dx x(1 + x2)-1

= (1 + x2)-1 d/dx x + x d/dx (1 + x2)-1

= (1 + x2)-1 + x -2x (1 + x2)-2

= (1 + x2)-2 (1 + x2 - 2x2)

= (1 - x2) / (1 + x2)2

• "over (1 + x) [all] squared": d/dx (x/(1+x)2)

d/dx(x/(1+x)2) = d/dx x(1 + x)-2

= (1 + x)-2 d/dx x + x d/dx (1 + x)-2

= (1 + x)-2 + x -2 (1 + x)-3

= (1 + x)-3 (1 + x - 2x)

= (1 - x)/(1 + x)3

If you prefer, you can use the quotient rule: d/dx (u/v) = (v du/dx - udv/dx) / v2