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Mathematical induction for differentiation

1

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

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

sinx

=

sin(x+\frac{1}{2}

n

\pi)

So to start off,

Let P(n) be

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

sinx

=

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,

Unparseable latex formula:

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

=

\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...

Reply 1

DFranklin

18

=

\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.

Reply 2

insparato

15

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

Reply 3

andrewlee89

OP

1

insparato

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

What does this mean?

Reply 4

DFranklin

18

andrewlee89

What does this mean?

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

Reply 5

andrewlee89

OP

1

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

DFranklin

18

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?

Reply 7

andrewlee89

OP

1

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.

Reply 8

lordcrusade9

1

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

andrewlee89

OP

1

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

Mathematical induction for differentiation

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