# Engineering Tables/Table of Derivatives

Table of Derivatives
${d \over dx} c = 0$
${d \over dx} x = 1$
${d \over dx} cx = c$
${d \over dx} |x| = {x \over |x|} = \sgn x,\qquad x \ne 0$
${d \over dx} x^c = cx^{c-1}$ where both xc and cxc-1 are defined.
${d \over dx} \left({1 \over x}\right) = {d \over dx} \left(x^{-1}\right) = -x^{-2} = -{1 \over x^2}$
${d \over dx} \left({1 \over x^c}\right) = {d \over dx} \left(x^{-c}\right) = -{c \over x^{c+1}}$
${d \over dx} \sqrt{x} = {d \over dx} x^{1\over 2} = {1 \over 2} x^{-{1\over 2}} = {1 \over 2 \sqrt{x}}$ x > 0
${d \over dx} c^x = {c^x \ln c}$ c > 0[/itex]
${d \over dx} e^x = e^x$
${d \over dx} \log_c x = {1 \over x \ln c}$ c > 0, $c \ne 1$
${d \over dx} \ln x = {1 \over x}$
${d \over dx} \sin x = \cos x$
${d \over dx} \cos x = -\sin x$
${d \over dx} \tan x = \sec^2 x$
${d \over dx} \sec x = \tan x \sec x$
${d \over dx} \cot x = -\csc^2 x$
${d \over dx} \csc x = -\csc x \cot x$
${d \over dx} \arcsin x = { 1 \over \sqrt{1 - x^2}}$
${d \over dx} \arccos x = {-1 \over \sqrt{1 - x^2}}$
${d \over dx} \arctan x = { 1 \over 1 + x^2}$
${d \over dx} \arcsec x = { 1 \over |x|\sqrt{x^2 - 1}}$
${d \over dx} \arccot x = {-1 \over 1 + x^2}$
${d \over dx} \arccsc x = {-1 \over |x|\sqrt{x^2 - 1}}$
${d \over dx} \sinh x = \cosh x$
${d \over dx} \cosh x = \sinh x$
${d \over dx} \tanh x = \mbox{sech}^2 x$
${d \over dx} \mbox{sech} x = - \tanh x \mbox{sech} x$
${d \over dx} \mbox{coth} x = - \mbox{csch}^2 x$
${d \over dx} \mbox{csch} x = - \mbox{coth} x \mbox{csch} x$
${d \over dx} \mbox{arcsinh} x = { 1 \over \sqrt{x^2 + 1}}$
${d \over dx} \mbox{arccosh} x = { 1 \over \sqrt{x^2 - 1}}$
${d \over dx} \mbox{arctanh} x = { 1 \over 1 - x^2}$
${d \over dx} \mbox{arcsech} x = { 1 \over x\sqrt{1 - x^2}}$
${d \over dx} \mbox{arccoth} x = { 1 \over 1 - x^2}$
${d \over dx} \mbox{arccsch} x = {-1 \over |x|\sqrt{1 + x^2}}$