How to Prove the Sum Rule

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Proving the sum rule will require you to find the derivative of the number in question. Prove the sum rule with help from an experienced math professional in this free video clip.

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Video Transcript

Hi, I'm Ryan Ault. I am a physicist and this is how to prove the sum rule. Now the sum rule is stated as the derivative with respect to x of some function f of x plus g of x is equal to the derivative with respect to x of f of x plus the derivative with respect to x of g of x. We can prove this by saying that the derivative with respect to x, some function is equal to the limit as d approaches zero of f of x plus d minus f of x over d. Now if we apply this rule to both terms up here, we find that d by dx of f of x plus g of x is simply equal to the limit as d approaches zero of f of x plus d minus f of x plus g of x plus d plus g of x and this is all over d. Now with the definition of limits we can associate distributively the limit with each term. And we find this simply goes to the limit as d approaches zero of f of x plus d minus f of x and you can see the second term is going to be the same thing except with gs. Now going back to our original definition of a limit, we see that it's actually equal to separate derivatives. So this can be concluded by saying that the sum rule is true because each term is a separate term which is a derivative. With respect to x of f of x and the derivative with respect to x of g of x. I'm Ryan Ault, and this is how to prove the sum rule.

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