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 $f_1(x)$, $f_2(x)$, $\cdots$, and $f_N(x)$. Prove that if $f(x)=f_1(x)+f_2(x)+\cdots+f_N(x)$ then $f'(x)=f'_1(x)+f'_2(x)+\cdots+f'_N(x)$.

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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