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Leibnitz Theorem

Hi, Im finding trouble trying to understand the Leibnitz theorem, can anyone help and just try simplifying it for me, Im just not getting it!:mad:

I got these 2 questions:
Use Leibnitz theorem to compute the 5th derivative of:
a) x^3cos(x)
b) x^2ln(x)

Thanks
This looks not fun. Is it the same as what this page is describing http://www.math.ohio-state.edu/~nevai/H16x/DOCUMENTS/leibniz_product_formula_H6.pdf

If so letting u(x) = x^3 and v(x) = cos(x), I'd then find the 1-5 th derivatives of each of them, then just stick it in that big sum formula.
Reply 2
Avatar for wazza_utd
wazza_utd
OP
1
Yeah we have to use the formula given at the top of the page but I dont really understand it.
I'll try write out the first few stages of the first one then.

You have x3cos⁡x x^3 \cos x so split this into two (easily differentiable) functions
Unparseable latex formula:

u(x) = x^

and v(x)=cos⁡x v(x) = \cos x . Find their 1st, 2nd, ..., 5th derivatives and just keep them somewhere handy.

Now do you understand the sum notation? This is what you'd do using the formula (I'm not going to write it out, I don't know the symbols)

Firstly, in the big sum, r = 0, and n = 5 because you want the 5th derivative.
So you get (5C0 is 5 choose 0, the binomial thingy)
5C0 x3d5(cosx)dx=−x3sin⁡x 5C0 \ x^3 \frac{d^5 (cos x)}{dx} = - x^3 \sin x
And now r = 1 so
5C1dx3dxd4(cosx)dx4=15x2cos⁡x 5C1 \frac{dx^3}{dx} \frac{d^4 (cos x)}{dx^4} = 15x^2 \cos x
And now do this for r= 2, 3, 4, and 5.
Finally add up all those bits because its a sum. Hopefully some stuff cancels out too.
Reply 4
Avatar for sebbie
sebbie
15
It's just the same as using the binomial theorem but substituting in derivatives, what part of it don;t you understand
Reply 5
Avatar for wazza_utd
wazza_utd
OP
1
Yeahh I think I get it now, I jus wasnt sure about the 'r' and 'n' term things. Thanks alot though
Reply 6
Avatar for wazza_utd
wazza_utd
OP
1
Sebbie I think I get it now, thanks
Reply 7
Avatar for MathsGirl
MathsGirl
12
if you were to find the odd derivatives how would you do that?

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