Successive Differentiation

$$ {d^n}/{dx^n} [a^x] = a^x (log_{e}a)^n $$

$$ {d^n}/{dx^n} [(ax + b)^m] = m(m-1)(m-2)…(m-n+1)a^n (ax+b)^{m-n} $$

$$ {d^n}/{dx^n} [{1}/{(ax+b)^m}] = {(-1)^n m(m+1)(m+2)…(m+n-1)a^n}/{(ax+b)^{m+n}} $$

Case: m > n
$$ {d^n}/{dx^n} [(ax+b)^m] = {m! a^{n} (ax+b)^{m-n}}/{(m-n)!} $$

Case: m = n
$$ {d^n}/{dx^n} [(ax+b)^n] = n! a^n$$

if a=1 and b = 0
$$ {d^n}/{dx^n} [x^n] = n! $$

Case: m is positive and m $$ {d^n}/{dx^n} [(ax+b)^m] = 0$$

Case: m = 1
$$ {d^n}/{dx^n} [{1}/{ax+b}] = {(-1)^n n! a^n}/{(ax+b)^{n+1}} $$

$$ {d^n}/{dx^n} [log(ax+b)] = {(-1)^{n-1} (n-1)! a^n}/{(ax+b)^n} $$

$$ {d^n}/{dx^n} [sin(bx+c)] = b^n sin(bx + c + {nπ}/{2}) $$

$$ {d^n}/{dx^n} [e^{ax} sin(bx+c)] = r^n e^{ax} sin(bx+c+nθ) $$

Leibnitz’s theorem:

$$ y_n = {^nC_0} u_n v + {^nC_1} u_{n-1} v_1 +…+ {^nC_n u v_n} $$