The Derivative of the Sum of a Finite Number of Differentiable Functions.

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Lesson Parent: 
Differentiation IV: Differentiation Formulas Everyone Must Know
Problem: 

Assume that we have N differentiable functions f1(x), f2(x), ⋯, and fN(x). Prove that if f(x)=f1(x)+f2(x)+⋯+fN(x) then f′(x)=f′1(x)+f′2(x)+⋯+f′N(x).

Answer: 

SAMPLE SOLUTION

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Solution: 

The proof will be done by induction. We have already proven that when N=2 that <Sign in to see all the formulas>.

Now we assume that it holds for k and then prove it for k+1. So now we assume

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and that <Sign in to see all the formulas> is differentiable. Therefore, <Sign in to see all the formulas> and since g(x) and <Sign in to see all the formulas> are both (by assumption) differentiable, then <Sign in to see all the formulas>. Now since <Sign in to see all the formulas> then <Sign in to see all the formulas>.

Since we have proven it works for k+1 when we assumed it works for k functions, we have shown that since it works for N=2 it will hold for any value of N.

Note that you may see this expressed as

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