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