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