Mathematical induction for differentiation Watch

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andrewlee89

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

$\frac{d^n}{dx^n}$

= $sin(x+\frac{1}{2}$ n $\pi)$

So to start off,

Let P(n) be $\frac{d^n}{dx^n}$

= $sin(x+\frac{1}{2}$ n $\pi)$

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

Suppose n=k,

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

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

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

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

DFranklin

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

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

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 $\sin(x+\frac{k\pi}{2})$ , just use the chain rule.

insparato

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

andrewlee89

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(Original post by insparato)
Yeah and remember sinx and cosx are the same graph just a translation apart.

What does this mean?

DFranklin

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(Original post by andrewlee89)
What does this mean?

cos x = sin(x+a) for a suitable choice of a.

andrewlee89

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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

DFranklin

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(Original post by 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 $\frac{k\pi}{2}$ to $\frac{(k+1)\pi}{2}$ ? Do you think it might possibly be by adding $\frac{\pi}{2}$ at some point?

andrewlee89

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

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

lordcrusade9

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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

andrewlee89

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#10

(Original post by 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

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