Math2.org Math Tables: Sum Rule

(Math)
(d/dx) [f(x) + g(x)] = (d/dx) f(x) + (d/dx) g(x)


Proof of (d/dx) [f(x) + g(x)] = (d/dx) f(x) + (d/dx) g(x) from the definition

We can use the definition of the derivative:

(d/dx) f(x) = lim
d-->0  		 f(x+d)-f(x)
d
Therefore, (d/dx) [f(x) + g(x)] can be written as such:
(d/dx) [f(x) + g(x)] =
lim
d-->0  		 [f(x+d)+g(x+d)] - [f(x)+g(x)]
d
= lim
   d-->0  		( [f(x+d)-f(x)]
d		 + [g(x+d)-g(x)]
d		)
= lim
   d-->0  		 f(x+d)-f(x)
d		 + lim
d-->0  		 g(x+d)-g(x)
d
= (d/dx) f(x) + (d/dx) g(x)