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Proof: Sum Rule
(Math | Calculus | Derivatives | Identities | Sum Rule)
(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
  
 
  

 
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