# Differentiation Formulas

In the formulas given below, it’s assumed that $$C$$, $$k$$ and $$n$$ are real numbers, $$m$$ is a natural number, $$f,g,u,v$$ are functions of the real variable $$x$$, and the base $$a$$ of the exponential and logarithmic functions satisfies the conditions $$a \gt 0, a \ne 1.$$ The domains of the functions and their graphs are given here.

 $$C’ = 0,\;C = \text{const}$$ $${\left[ {f\left( x \right) \pm g\left( x \right)} \right]^\prime } = {f’\left( x \right) \pm g’\left( x \right)}$$ $${\left( {kf\left( x \right)} \right)^\prime } = {kf’\left( x \right),\;}\kern-0.3pt{k = \text{const}}$$ $${\left( {uv} \right)^\prime } = {u’v + uv’}$$ $${\left( {\large\frac{u}{v}\normalsize} \right)^\prime } = \large\frac{{u’v – uv’}}{{{v^2}}}\normalsize$$ $${\left( {\large\frac{1}{{f\left( x \right)}}\normalsize} \right)^\prime } = – {\large\frac{{f’\left( x \right)}}{{{f^2}\left( x \right)}}\normalsize}$$ $${\left( {uv} \right)^{\prime\prime}} = {u^{\prime\prime}v + 2u’v’ + uv^{\prime\prime}}$$ $${\left( {uv} \right)^{\prime\prime\prime}} = {u^{\prime\prime\prime}v + 3u^{\prime\prime}v’ }+{ 3u’v^{\prime\prime} }+{ uv^{\prime\prime\prime}}$$ $${\left( {u + v} \right)^{\left( n \right)}} = {u^{\left( n \right)}} + {v^{\left( n \right)}}$$ $${\left( {Cu} \right)^{\left( n \right)}} = C{u^{\left( n \right)}}$$ $${\left( {uv} \right)^{\left( n \right)}} = {{u^{\left( n \right)}}v + n{u^{\left( {n – 1} \right)}}v’ }+{ {\large\frac{{n\left( {n – 1} \right)}}{{1 \cdot 2}}\normalsize}{u^{\left( {n – 2} \right)}}v^{\prime\prime}} + { \ldots }+{ u{v^{\left( n \right)}}}$$ $$x’ = 1$$ $${\left( {{x^2}} \right)^\prime } = 2x$$ $${\left( {{x^n}} \right)^\prime } = n{x^{n – 1}}$$ $${\left( {\large\frac{1}{x}\normalsize} \right)^\prime } = – \large\frac{1}{{{x^2}}}\normalsize$$ $${\left( {\large\frac{1}{{{x^n}}}\normalsize} \right)^\prime } = – \large\frac{n}{{{x^{n + 1}}}\normalsize}$$ $${\left( {\sqrt x } \right)^\prime } = {\large\frac{1}{{2\sqrt x }}\normalsize}$$ $${\left( {\sqrt[\large m\normalsize]{x}} \right)^\prime } = \large\frac{1}{{m\sqrt[m]{{{x^{m – 1}}}}}}\normalsize$$ $${\left( {{a^x}} \right)^\prime } = {a^x}\ln a$$ $${\left( {{e^x}} \right)^\prime } = {e^x}$$ $${\left( {{{\log }_a}x} \right)^\prime } = \large\frac{1}{{x\ln a}}\normalsize$$ $${\left( {\ln x} \right)^\prime } = \large\frac{1}{x}\normalsize$$ $${\left( {\sin x} \right)^\prime } = \cos x$$ $${\left( {\cos x} \right)^\prime } = – \sin x$$ $${\left( {\tan x} \right)^\prime } = {\large\frac{1}{{{{\cos }^2}x}}\normalsize} = {\sec ^2}x$$ $${\left( {\cot x} \right)^\prime } = – {\large\frac{1}{{{\sin^2}x}}\normalsize} = – {\csc ^2}x$$ $${\left( {\sec x} \right)^\prime } = \tan x\sec x$$ $${\left( {\csc x} \right)^\prime } = – \cot x\csc x$$ $${\left( {\arcsin x} \right)^\prime } = {\large\frac{1}{{\sqrt {1 – {x^2}} }}\normalsize}$$ $${\left( {\arccos x} \right)^\prime } = -\large\frac{1}{{\sqrt {1 – {x^2}} }}\normalsize$$ $${\left( {\arctan x} \right)^\prime } = {\large\frac{1}{{1 + {x^2}}}\normalsize}$$ $${\left( {\text{arccot}\,x} \right)^\prime } = -\large\frac{1}{{1 + {x^2}}}\normalsize$$ $${\left( {\text{arcsec}\,x} \right)^\prime } = {\large\frac{1}{{\left| x \right|\sqrt {{x^2} – 1} }}\normalsize}$$ $${\left( {\text{arccsc}\,x} \right)^\prime } = -\large\frac{1}{{\left| x \right|\sqrt {{x^2} – 1} }}\normalsize$$ $${\left( {\sinh x} \right)^\prime } = \cosh x$$ $${\left( {\cosh x} \right)^\prime } = \sinh x$$ $${\left( {\tanh x} \right)^\prime } = {\text{sech}^2}x$$ $${\left( {\text{coth}\,x} \right)^\prime } = -{\text{csch}^2}x$$ $${\left( {\text{sech}\,x} \right)^\prime } = – \text{sech}\,x\tanh x$$ $${\left( {\text{csch}\,x} \right)^\prime } = – \text{csch}\,x\,\text{coth}\,x$$ $${\left( {\text{arcsinh}\,x} \right)^\prime } = \large\frac{1}{{\sqrt {{x^2} + 1} }}\normalsize$$ $${\left( {\text{arccosh}\,x} \right)^\prime } = \large\frac{1}{{\sqrt {{x^2} – 1} }}\normalsize$$ $${\left( {\text{arctanh}\,x} \right)^\prime } = \large\frac{1}{{1 – {x^2}}}\normalsize$$
 $$C’ = 0,\;C = \text{const}$$ $${\left[ {f\left( x \right) \pm g\left( x \right)} \right]^\prime } = {f’\left( x \right) \pm g’\left( x \right)}$$ $${\left( {kf\left( x \right)} \right)^\prime } = {kf’\left( x \right),\;}\kern-0.3pt{k = \text{const}}$$ $${\left( {uv} \right)^\prime } = {u’v + uv’}$$ $${\left( {\large\frac{u}{v}\normalsize} \right)^\prime } = \large\frac{{u’v – uv’}}{{{v^2}}}\normalsize$$ $${\left( {\large\frac{1}{{f\left( x \right)}}\normalsize} \right)^\prime } = – {\large\frac{{f’\left( x \right)}}{{{f^2}\left( x \right)}}\normalsize}$$ $${\left( {uv} \right)^{\prime\prime}} = {u^{\prime\prime}v + 2u’v’ + uv^{\prime\prime}}$$ $${\left( {uv} \right)^{\prime\prime\prime}} = {u^{\prime\prime\prime}v + 3u^{\prime\prime}v’ }+{ 3u’v^{\prime\prime} }+{ uv^{\prime\prime\prime}}$$ $${\left( {u + v} \right)^{\left( n \right)}} = {u^{\left( n \right)}} + {v^{\left( n \right)}}$$ $${\left( {Cu} \right)^{\left( n \right)}} = C{u^{\left( n \right)}}$$ $${\left( {uv} \right)^{\left( n \right)}} = {{u^{\left( n \right)}}v + n{u^{\left( {n – 1} \right)}}v’ }+{ {\large\frac{{n\left( {n – 1} \right)}}{{1 \cdot 2}}\normalsize}{u^{\left( {n – 2} \right)}}v^{\prime\prime}} + { \ldots }+{ u{v^{\left( n \right)}}}$$ $$x’ = 1$$ $${\left( {{x^2}} \right)^\prime } = 2x$$ $${\left( {{x^n}} \right)^\prime } = n{x^{n – 1}}$$ $${\left( {\large\frac{1}{x}\normalsize} \right)^\prime } = – \large\frac{1}{{{x^2}}}\normalsize$$ $${\left( {\large\frac{1}{{{x^n}}}\normalsize} \right)^\prime } = – \large\frac{n}{{{x^{n + 1}}}\normalsize}$$ $${\left( {\sqrt x } \right)^\prime } = {\large\frac{1}{{2\sqrt x }}\normalsize}$$ $${\left( {\sqrt[\large m\normalsize]{x}} \right)^\prime } = \large\frac{1}{{m\sqrt[m]{{{x^{m – 1}}}}}}\normalsize$$ $${\left( {{a^x}} \right)^\prime } = {a^x}\ln a$$ $${\left( {{e^x}} \right)^\prime } = {e^x}$$ $${\left( {{{\log }_a}x} \right)^\prime } = \large\frac{1}{{x\ln a}}\normalsize$$ $${\left( {\ln x} \right)^\prime } = \large\frac{1}{x}\normalsize$$ $${\left( {\sin x} \right)^\prime } = \cos x$$ $${\left( {\cos x} \right)^\prime } = – \sin x$$ $${\left( {\tan x} \right)^\prime } = {\large\frac{1}{{{{\cos }^2}x}}\normalsize} = {\sec ^2}x$$ $${\left( {\cot x} \right)^\prime } = – {\large\frac{1}{{{\sin^2}x}}\normalsize} = – {\csc ^2}x$$ $${\left( {\sec x} \right)^\prime } = \tan x\sec x$$ $${\left( {\csc x} \right)^\prime } = – \cot x\csc x$$ $${\left( {\arcsin x} \right)^\prime } = {\large\frac{1}{{\sqrt {1 – {x^2}} }}\normalsize}$$ $${\left( {\arccos x} \right)^\prime } = -\large\frac{1}{{\sqrt {1 – {x^2}} }}\normalsize$$ $${\left( {\arctan x} \right)^\prime } = {\large\frac{1}{{1 + {x^2}}}\normalsize}$$ $${\left( {\text{arccot}\,x} \right)^\prime } = -\large\frac{1}{{1 + {x^2}}}\normalsize$$ $${\left( {\text{arcsec}\,x} \right)^\prime } = {\large\frac{1}{{\left| x \right|\sqrt {{x^2} – 1} }}\normalsize}$$ $${\left( {\text{arccsc}\,x} \right)^\prime } = -\large\frac{1}{{\left| x \right|\sqrt {{x^2} – 1} }}\normalsize$$ $${\left( {\sinh x} \right)^\prime } = \cosh x$$ $${\left( {\cosh x} \right)^\prime } = \sinh x$$ $${\left( {\tanh x} \right)^\prime } = {\text{sech}^2}x$$ $${\left( {\text{coth}\,x} \right)^\prime } = -{\text{csch}^2}x$$ $${\left( {\text{sech}\,x} \right)^\prime } = – \text{sech}\,x\tanh x$$ $${\left( {\text{csch}\,x} \right)^\prime } = – \text{csch}\,x\,\text{coth}\,x$$ $${\left( {\text{arcsinh}\,x} \right)^\prime } = \large\frac{1}{{\sqrt {{x^2} + 1} }}\normalsize$$ $${\left( {\text{arccosh}\,x} \right)^\prime } = \large\frac{1}{{\sqrt {{x^2} – 1} }}\normalsize$$ $${\left( {\text{arctanh}\,x} \right)^\prime } = \large\frac{1}{{1 – {x^2}}}\normalsize$$