Proof by Induction of the Power Rule of Differentiation

[tinylights]

Homework Statement

Okay, the concept here is to use induction to prove that for n, (f₁ x f₂ x ... x f_n-1 x f_n)' = (f'₁ x f₂ x ... x f_n) + (f₁ x f'₂ x ... x f_n) + ... + (f₁ x f₂ x ... x f'_n).

2. Homework Equations / 3. The Attempt at a Solution

I solved the initial step, which was quite easy. I started to set up the inductive step, by stating that:

(f₁ x f₂ ... x f_n x f_n+1)' = (f'₁ x f₂ x ... x f_n+1) + (f₁ x f'₂ x ... x f_n+1) + ... + (f₁ x f₂ x ... x f'_n+1).

And I do understand how induction works - I know I am supposed to plug in what I have for the "n" equation into part of my "n+1" equation and find equality. I just don't know HOW I'm supposed to do that for some reason. I think it's the derivatives throwing me off because in the last few problems that I did, it was fine. Please help me simplify?

 

[Dick]

Homework Statement

Okay, the concept here is to use induction to prove that for n, (f₁ x f₂ x ... x f_n-1 x f_n)' = (f'₁ x f₂ x ... x f_n) + (f₁ x f'₂ x ... x f_n) + ... + (f₁ x f₂ x ... x f'_n).

2. Homework Equations / 3. The Attempt at a Solution

I solved the initial step, which was quite easy. I started to set up the inductive step, by stating that:

(f₁ x f₂ ... x f_n x f_n+1)' = (f'₁ x f₂ x ... x f_n+1) + (f₁ x f'₂ x ... x f_n+1) + ... + (f₁ x f₂ x ... x f'_n+1).

And I do understand how induction works - I know I am supposed to plug in what I have for the "n" equation into part of my "n+1" equation and find equality. I just don't know HOW I'm supposed to do that for some reason. I think it's the derivatives throwing me off because in the last few problems that I did, it was fine. Please help me simplify?

(f1 x f2 ... x fn x fn+1)=(f1 x f2 ... x fn) x fn+1. Use the usual two factor product rule on that.

 

[tinylights]

Okay, that makes it simpler. I didn't know if I was "allowed" to assume that, or if I had to go from the ground up. I'll double-check with my professor but for now I'll work it out that way. Thanks!

 

[Dick]

Okay, that makes it simpler. I didn't know if I was "allowed" to assume that, or if I had to go from the ground up. I'll double-check with my professor but for now I'll work it out that way. Thanks!

I didn't "assume" anything. (fg)'=f'g+fg' is the base case for your induction. It's the usual product rule. Sure, you can assume that. And (f1 x f2 ... x fn x fn+1)=(f1 x f2 ... x fn) x fn+1 is just using the associative rule for multiplication. I think you are pretty safe in using that!

 

[HallsofIvy]

You titled this "power rule" but talk about the "product rule". Which are you trying to prove?