22. Differentiation -2

Leibnitz Theorem and nth derivative

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

Then

(uv)_n = u_n(x)v(x)+ⁿC₁u_n-1(x)v₁(x)+...+ⁿC_ku_n-k(x)v_k(x) +...+ⁿC_nu(x)v_n(x)


where u_k(x)= d^k(u(x))/dx^k (k-th derivative of the function u(x), 1≤k≤n


n-derivatives of some elementary functions


dⁿ(x^m)/dxⁿ = (m!x^m-n)/(m-n)! if n≤m

and
dⁿ(x^m)/dxⁿ = 0 if n>m.


dⁿ(sin x)/dxⁿ = sin (x + nπ/2)

dⁿ(cos x)/dxⁿ = cos (x + nπ/2)

dⁿ(e^mx)/dxⁿ = mⁿe^mx