Help with mathematical induction

[decemberflower]

Hello,

I have my Math HL exams in 5 weeks (and all other exams too naturally). Anyway, I am practicing right now and I just encountered this problem that I cannot solve. Looking at the answer, I understand the concept but not the actual working. I'd be very thankful if anyone could look at it and possibly offer an intermediate step.

I attached problem & answer. So, first you assume the claim works also for n=k. Yes. Then, to prove it for n=k+1, they take the expression for k from the assumption and differentiate it again. I get that, too. But I don't understand the working of the differentiation [from 2nd line of d^(k+1)y/ dx(k+1)on]. Do they use the product rule? Still, I can't follow them . Well, perhaps someone has a nice and easy explanation.

Thanks.

Viola

[Jennifer]

They have taken (-1)^k+1 out as a sort of common factor and differentiated (using the product rule) the rest of it. This is because (-1)^k+1 acts as a constant, acting as a 2 would in 2x^3 (i.e. remains the same - not differentiated).

In the last two lines they have just gathered like terms etc, and taken a -1 out as a common factor, turning the (-1)^k+1 into a -1 x (-1)^k+1 and hence (-1)^k+2.

Not sure if that made any sense, but just remember that k is not something you need to differentiate, as you are finding dy/dx and not dy/dk.

[decemberflower]

It did. Thank you .

Although I like math I'm just about ready to go crazy these days.. it's just SO MUCH that we have to know at once. Anyways... it'll work out somehow.

[Survival Robot]

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