Sum rule in differentiation

The sum rule is in mathematics one of the basic rules of differential calculus. It says that the sum of two differentiable functions is differentiable and again that such a sum of functions can be differentiated term by term.

Rule

And the functions are defined in a common interval that contains the point. At this point, both functions are differentiable. Then the function

Differentiable at the point, and it is

Example

The functions

Are differentiable with derivative functions

Therefore, the function

On differentiable with the derivative function

Conclusions

• Difference rule considering the difference of functions and which are differentiated, resulting from the sum rule and the control factor, that is differentiable, and is for the discharge.
• Together with the factor usually results: Are in differentiable functions and are real constants, then the linear combination is again in differentiable with ( term-wise differentiated ) derivative function.

• It follows that the differentiable functions ( at a given interval), forming a real vector space, and the differentiation is a linear map of this vector space in the vector space of all functions.