Sum Rule

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Proof: Sum Rule

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

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

f(x) + (d/dx)

g(x)

Proof of

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

f(x) + (d/dx)

g(x) from the definition

We can use the definition of the derivative:

f(x) = lim
^d-->0 f(x+d)-f(x)
d

Therefore, (d/dx)

[f(x) + g(x)] can be written as such:

[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