Differentiate

Possible derivation:
d/dx((x f''(x))/(2 + x^2))
Use the quotient rule, d/dx(u/v) = (v ( du)/( dx) - u ( dv)/( dx))/v^2, where u = x f''(x) and v = x^2 + 2:
 = ((2 + x^2) (d/dx(x f''(x))) - x (d/dx(2 + x^2)) f''(x))/(2 + x^2)^2

Use the product rule, d/dx(u v) = v ( du)/( dx) + u ( dv)/( dx), where u = x and v = f''(x):
 = ((2 + x^2) x (d/dx(f''(x))) + (d/dx(x)) f''(x) - x (d/dx(2 + x^2)) f''(x))/(2 + x^2)^2
The derivative of f''(x) is f^(3)(x):
 = (-x (d/dx(2 + x^2)) f''(x) + (2 + x^2) (f^(3)(x) x + (d/dx(x)) f''(x)))/(2 + x^2)^2

Differentiate the sum term by term:
 = (-x f''(x) d/dx(2) + d/dx(x^2) + (2 + x^2) ((d/dx(x)) f''(x) + x f^(3)(x)))/(2 + x^2)^2
The derivative of 2 is zero:
 = (-x (d/dx(x^2) + 0) f''(x) + (2 + x^2) ((d/dx(x)) f''(x) + x f^(3)(x)))/(2 + x^2)^2

Simplify the expression:
 = (-x (d/dx(x^2)) f''(x) + (2 + x^2) ((d/dx(x)) f''(x) + x f^(3)(x)))/(2 + x^2)^2
Use the power rule, d/dx(x^n) = n x^(n - 1), where n = 2: d/dx(x^2) = 2 x:
 = (-x f''(x) 2 x + (2 + x^2) ((d/dx(x)) f''(x) + x f^(3)(x)))/(2 + x^2)^2

Simplify the expression:
 = (-2 x^2 f''(x) + (2 + x^2) ((d/dx(x)) f''(x) + x f^(3)(x)))/(2 + x^2)^2
The derivative of x is 1:
Answer: | = (-2 x^2 f''(x) + (2 + x^2) (1 f''(x) + x f^(3)(x)))/(2 + x^2)^2

"What is the f''(x) of (x/(x2+2))"

.