(Here f(x) is x and 0 is its power, derivative of x is 1 which is also exist in front of x. So apply the rule.)

0 + 1
x
= C ------- + A
0 + 1
1
x
= C ----- + A
1
= Cx + A

(Here A is constant. We can also take C as constant but a constant C is already exit in question. So the new constant may or may not equal to C. If equal to C then we can take common C then answer may become C(x + 1).)