Shortcuts for Differentiation

While the limit definition of the derivative can allow differentiation, it is often tedious and the shortcuts for differentiation can save time.  Some shortcuts include:

d/dx(C)=0
d/dx(u^n) = nu^(n-1) du/dx
d/dx(sin(u)) = cos(u) du/dx
d/dx(cos(u)) = -sin(u) du/dx
d/dx(tan(u)) = sec(u)^2 du/dx
d/dx(cot(u)) = -csc(u)^2 du/dx
d/dx(sec(u)) = sec(u)tan(u) du/dx
d/dx(csc(u)) = -csc(u)cot(u) du/dx
d/dx(logau) = (logeu)/u du/dx
d/dx(ln(u)) = 1/u du/dx
d/dx(a^u) = a^u ln(a) du/dx
d/dx(e^u) = e^u du/dx
d/dx(arcsin(u)) = 1/(1-u^2)^.5 du/dx
d/dx(arccos(u)) = -1/(1-u^2)^.5 du/dx
d/dx(arctan(u)) = 1/(1+u^2) du/dx
d/dx(arccot(u)) = -1/(1+u^2) du/dx
d/dx(arcsec(u)) = +-1/(u(u^2-1)^.5) du/dx (+ if u > 1 or - if u < -1)
d/dx(arccsc(u)) = +-1/(u(u^2-1)^.5) du/dx (- if u > 1 or + if u < -1)
d/dx(sinh(u)) = cosh(u) du/dx
d/dx(cosh(u)) = sinh(u) du/dx
d/dx(tanh(u)) = sech(u)^2 du/dx
d/dx(coth(u)) = -csch(u)^2 du/dx
d/dx(sech(u)) = -sech(u)tanh(u) du/dx
d/dx(csch(u)) = -csch(u)coth(u) du/dx
d/dx(arcsinh(u)) = 1/(1+u^2)^.5 du/dx
d/dx(arccosh(u)) = 1/(u^2-1)^.5 du/dx
d/dx(arctanh(u)) = 1/(1-u^2) du/dx
d/dx(arccoth(u)) = 1/(1-u^2) du/dx
d/dx(arcsech(u)) = 1/(u(1-u^2)^.5) du/dx
d/dx(arccsch(u)) = -1/(u(u^2+1)) du/dx