Functions: \(f\), \(y\), \(u\), \(v\)

Argument (independent variable): \(x\)

Argument (independent variable): \(x\)

Natural numbers: \(n\), \(m\)

Real number: \(a\)

Real number: \(a\)

- Notation of the second order derivative

\(f^{\prime\prime} = \left( {f’} \right)’ = {\left( {{\large\frac{{dy}}{{dx}}}\normalsize} \right)^\prime } =\) \({\large\frac{d}{{dx}}\normalsize}\left( {{\large\frac{{dy}}{{dx}}}\normalsize} \right) =\) \({\large\frac{{{d^2}y}}{{d{x^2}}}\normalsize}\) - Notation of the derivative of order \(n\)

\({f^{\left( n \right)}} = {\large\frac{{{d^n}y}}{{d{x^n}}}\normalsize} = {y^{\left( n \right)}} =\) \({\left( {{f^{\left( {n – 1} \right)}}} \right)^\prime }\) - \(N\)th order derivative of the sum of functions

\({\left( {u + v} \right)^{\left( n \right)}} =\) \({u^{\left( n \right)}} + {v^{\left( n \right)}}\) - \(N\)th order derivative of the difference of functions

\({\left( {u – v} \right)^{\left( n \right)}} =\) \({u^{\left( n \right)}} – {v^{\left( n \right)}}\) - Leibniz formula

\({\left( {uv} \right)^{\left( n \right)}} = {u^{\left( n \right)}}v \,+\) \(n{u^{\left( {n – 1} \right)}}v^{\prime} \,+\) \({\large\frac{{n\left( {n – 1} \right)}}{{1 \cdot 2}}\normalsize} {u^{\left( {n – 2} \right)}}v^{\prime\prime} + \ldots\) \(+\, u{v^{\left( n \right)}},\) \({\left( {uv} \right)}^{\prime\prime} = {u^{\prime\prime}}v + 2u’v’ \,+\) \(uv^{\prime\prime},\) \({\left( {uv} \right)}^{\prime\prime\prime} = {u^{\prime\prime\prime}}v \,+\) \(3u^{\prime\prime}v’ \,+\) \(3u’v^{\prime\prime} + uv^{\prime\prime\prime}.\) - \(N\)th order derivative of the power function

\({\left( {{x^m}} \right)^{\left( n \right)}} = {\large\frac{{m!}}{{\left( {m – n} \right)!}}\normalsize} {x^{m – n}}\) - \(N\)th order derivative of the function \(y = x^n\)

\({\left( {{x^n}} \right)^{\left( n \right)}} = n!\) - \(N\)th order derivative of the logarithmic function

\({\left( {{{\log }_a}x} \right)^{\left( n \right)}} =\) \( {\large\frac{{{{\left( { – 1} \right)}^{n – 1}}\left( {n – 1} \right)!}}{{{x^n}\ln a}}\normalsize}\) - \(N\)th order derivative of the natural logarithm

\({\left( {\ln x} \right)^{\left( n \right)}} = {\large\frac{{{{\left( { – 1} \right)}^{n – 1}}\left( {n – 1} \right)!}}{{x^n}}\normalsize}\) - \(N\)th order derivative of the exponential function with base \(a\)

\({\left( {{a^x}} \right)^{\left( n \right)}} = {a^x}{\ln ^n}a\) - \(N\)th order derivative of the exponential function with base \(e\)

\({\left( {{e^x}} \right)^{\left( n \right)}} = {e^x}\) - \(N\)th order derivative of the function \(y = a^{mx}\)

\({\left( {{a^{mx}}} \right)^{\left( n \right)}} = {m^n}{a^{mx}}{\ln ^n}a\) - \(N\)th order derivative of the sine function

\({\left( {\sin x} \right)^{\left( n \right)}} =\) \( \sin \left( {x + {\large\frac{{\pi n}}{2}}\normalsize} \right)\) - \(N\)th order derivative of the cosine function

\({\left( {\cos x} \right)^{\left( n \right)}} =\) \( \cos \left( {x + {\large\frac{{\pi n}}{2}}\normalsize} \right)\)