Sunday, May 25, 2008

22. Differentiation -2

Leibnitz Theorem and nth derivative

if u(x) and v(x)are functions possessing derivatives up to nth order.


(uv)n = un(x)v(x)+nC1un-1(x)v1(x)+...+nCkun-k(x)vk(x) +...+nCnu(x)vn(x)

where uk(x)= dk(u(x))/dxk (k-th derivative of the function u(x), 1≤k≤n

n-derivatives of some elementary functions

dn(xm)/dxn = (m!xm-n)/(m-n)! if n≤m

dn(xm)/dxn = 0 if n>m.

dn(sin x)/dxn = sin (x + nπ/2)

dn(cos x)/dxn = cos (x + nπ/2)

dn(emx)/dxn = mnemx

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