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(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)


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