Leibniz's Theorem: The nth derivative of a product of two functions u and v: (uv)^(n) = sum(r=0 to n) C(n,r) * u^(n-r) * v^(r)

where u^(k) denotes the kth derivative of u, and C(n,r) = n!/(r!(n-r)!).

Expanding: (uv)^(n) = u^(n)v + nu^(n-1)*v' + C(n,2)*u^(n-2)v'' + ... + uv^(n)

This is analogous to the binomial expansion but with derivatives.

Common application: Find the nth derivative of $x^2$ * $e^x$. Let u = $e^x$ (all derivatives are $e^x$) and v = $x^2$ (v' = 2x, v'' = 2, v''' = 0...). ($x^2$$e^x$)^(n) = C(n,0)$e^x$$x^2$ + C(n,1)$e^x$2x + C(n,2)$e^x$*2 = $e^x$ * [$x^2$ + 2nx + n(n-1)]

Useful nth derivatives:

• (e^(ax))^(n) = $a^n$ * e^(ax)
• ($x^m$)^(n) = m!/(m-n)! * x^(m-n) for n <= m; = 0 for n > m
• (sin(ax+b))^(n) = $a^n$ * sin(ax + b + n*pi/2)
• (cos(ax+b))^(n) = $a^n$ * cos(ax + b + n*pi/2)
• (ln x)^(n) = (-1)^(n-1) * (n-1)! / $x^n$