Derivatives

(1) $d/{dx} x^n = n x^{n-1}$

(2) $d/{dx}$ k (constant) = 0

(3) $d/{dx}$ sinx = cosx

(4) $d/{dx}$ cosx = -sinx

(5) $d/{dx} tanx = sec^2 x$

(6) $d/{dx} cotx = – cosec^2 x$

(7) $d/{dx} secx = secx tanx$

(8) $d/{dx} cosecx = – cosecx cotx$

(9)$ d/{dx} a^x = a^x log a $

(10) $d/{dx} logx = 1/x$

(11) $d/{dx} e^x = e^x$

(12) $d/{dx} sin^{-1}x = {1}/{√{1 – x^2}}$

(13) $d/{dx} cos^{-1}x = – {1}/{√{1 – x^2}}$

(14) $d/{dx} tan^{-1}x = 1/{1+x^2}$

(15) $d/{dx} cot^{-1}x = – {1}/{1 + x^2}$

(16) $d/{dx} sec^{-1}x = 1/{x√{x^2 – 1}}$

(17) $d/{dx} cosec^{-1}x = – {1}/{x√{x^2 – 1}}$

(18) $d/{dx} uv = uv’ + vu’$

(19) $d/{dx} u/v = {vu’ – uv’}/{v^2}$

(20) ${dy}/{dx} = y(1 + logx)$

(21) ${dy}/{dx} = {dy}/{dt}$ x ${dt}/{dx}$

(22) $tan^{-1}x + tan^{-1}y = tan^{-1}({x+y}/{1 – xy})$ (xy < 1)

(23) $tan^{-1}x + tan^{-1}y = π + tan^{-1}({x+y}/{1-xy})$ (xy > 1)

(24) $tan^{-1}x + tan^{-1}y = tan^{-1} ({x-y}/{1+xy})$

(25) $2 tan^{-1}x = sin^{-1}({2x}/{1 + x^2})$ |x| ≤ 1

(26) $2 tan^{-1}x = cos^{-1}({1 – x^2}/{1 + x^2})$ x ≥ 0

(27) $2 tan^{-1}x = tan^{-1}({2x}/{1 – x^2})$ -1 < x < 1