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Mathematical induction for differentiation

Use mathematical induction to prove that for all positive integers n,

dndxn\frac{d^n}{dx^n}sinxsinx = sin(x+12sin(x+\frac{1}{2}nπ)\pi)

So to start off,

Let P(n) be dndxn\frac{d^n}{dx^n}sinxsinx = sin(x+12sin(x+\frac{1}{2}nπ)\pi)

Basis case, when n=1, ddx(sinx)=cosx\frac{d}{dx}(sinx) = cosx
sin(x+π2)=cosxsin(x+\frac{\pi}{2})=cosx

Suppose n=k,

Unparseable latex formula:

\frac{d^k^+^1}{dx^k^+^1}(sinx) = \frac{d}{dx}(\frac{d^k}{dx^k}sinx)



= ddx(sin(x+12kπ)\frac{d}{dx}(sin(x+\frac{1}{2}k\pi)

= ddx(sinxcos12kπ+cosxsin12kπ)\frac{d}{dx}(sinxcos\frac{1}{2}k\pi + cosxsin\frac{1}{2}k\pi)

Can anyone help me go from here? I'm nt sure how to clear off the k's in the angle right now...

Reply 1


= ddx(sin(x+12kπ)\frac{d}{dx}(sin(x+\frac{1}{2}k\pi)

= ddx(sinxcos12kπ+cosxsin12kπ)\frac{d}{dx}(sinxcos\frac{1}{2}k\pi + cosxsin\frac{1}{2}k\pi)

Can anyone help me go from here? I'm nt sure how to clear off the k's in the angle right now...Instead of expanding sin(x+kπ2)\sin(x+\frac{k\pi}{2}), just use the chain rule.

Reply 2

Yeah and remember sinx and cosx are the same graph just a translation apart.

Reply 3

insparato
Yeah and remember sinx and cosx are the same graph just a translation apart.

What does this mean?

Reply 4

andrewlee89
What does this mean?
cos x = sin(x+a) for a suitable choice of a.

Reply 5

Oh.. Well that's true but this is proof by induction I need to get the whole equation into the k+1 format for proof

Reply 6

andrewlee89
Oh.. Well that's true but this is proof by induction I need to get the whole equation into the k+1 format for proof
Yes, we know it's proof by induction. How do you think you are going to go from kπ2\frac{k\pi}{2} to (k+1)π2\frac{(k+1)\pi}{2}? Do you think it might possibly be by adding π2\frac{\pi}{2} at some point?

Reply 7

DFranklin
Yes, we know it's proof by induction. How do you think you are going to go from kπ2\frac{k\pi}{2} to (k+1)π2\frac{(k+1)\pi}{2}? Do you think it might possibly be by adding π2\frac{\pi}{2} at some point?

True :P Thx for the help. Will continue at it.

Reply 8

u got it yet?

cos x = sin (x+pi/2) or sin x = cos (x-pi/2) is the thing u need. it is actually a very short proof

Reply 9

lordcrusade9
u got it yet?

cos x = sin (x+pi/2) or sin x = cos (x-pi/2) is the thing u need. it is actually a very short proof

Got it thx guys. U all rock. At various different times lol