Formulas and Tables

Calculus

Table of Higher Order Derivatives

Functions: \(f\), \(y\), \(u\), \(v\)
Argument (independent variable): \(x\)

Natural numbers: \(n\), \(m\)
Real number: \(a\)

  1. Notation of the second order derivative
    \(f^{\prime\prime} = \left( {f’} \right)’ = {\left( {{\large\frac{{dy}}{{dx}}}\normalsize} \right)^\prime } =\) \({\large\frac{d}{{dx}}\normalsize}\left( {{\large\frac{{dy}}{{dx}}}\normalsize} \right) =\) \({\large\frac{{{d^2}y}}{{d{x^2}}}\normalsize}\)
  2. Notation of the derivative of order \(n\)
    \({f^{\left( n \right)}} = {\large\frac{{{d^n}y}}{{d{x^n}}}\normalsize} = {y^{\left( n \right)}} =\) \({\left( {{f^{\left( {n – 1} \right)}}} \right)^\prime }\)
  3. \(N\)th order derivative of the sum of functions
    \({\left( {u + v} \right)^{\left( n \right)}} =\) \({u^{\left( n \right)}} + {v^{\left( n \right)}}\)
  4. \(N\)th order derivative of the difference of functions
    \({\left( {u – v} \right)^{\left( n \right)}} =\) \({u^{\left( n \right)}} – {v^{\left( n \right)}}\)
  5. Leibniz formula
    \({\left( {uv} \right)^{\left( n \right)}} = {u^{\left( n \right)}}v \,+\) \(n{u^{\left( {n – 1} \right)}}v^{\prime} \,+\) \({\large\frac{{n\left( {n – 1} \right)}}{{1 \cdot 2}}\normalsize} {u^{\left( {n – 2} \right)}}v^{\prime\prime} + \ldots\) \(+\, u{v^{\left( n \right)}},\)
    \({\left( {uv} \right)}^{\prime\prime} = {u^{\prime\prime}}v + 2u’v’ \,+\) \(uv^{\prime\prime},\)
    \({\left( {uv} \right)}^{\prime\prime\prime} = {u^{\prime\prime\prime}}v \,+\) \(3u^{\prime\prime}v’ \,+\) \(3u’v^{\prime\prime} + uv^{\prime\prime\prime}.\)
  6. \(N\)th order derivative of the power function
    \({\left( {{x^m}} \right)^{\left( n \right)}} = {\large\frac{{m!}}{{\left( {m – n} \right)!}}\normalsize} {x^{m – n}}\)
  7. \(N\)th order derivative of the function \(y = x^n\)
    \({\left( {{x^n}} \right)^{\left( n \right)}} = n!\)
  8. \(N\)th order derivative of the logarithmic function
    \({\left( {{{\log }_a}x} \right)^{\left( n \right)}} =\) \( {\large\frac{{{{\left( { – 1} \right)}^{n – 1}}\left( {n – 1} \right)!}}{{{x^n}\ln a}}\normalsize}\)
  9. \(N\)th order derivative of the natural logarithm
    \({\left( {\ln x} \right)^{\left( n \right)}} = {\large\frac{{{{\left( { – 1} \right)}^{n – 1}}\left( {n – 1} \right)!}}{{x^n}}\normalsize}\)
  10. \(N\)th order derivative of the exponential function with base \(a\)
    \({\left( {{a^x}} \right)^{\left( n \right)}} = {a^x}{\ln ^n}a\)
  11. \(N\)th order derivative of the exponential function with base \(e\)
    \({\left( {{e^x}} \right)^{\left( n \right)}} = {e^x}\)
  12. \(N\)th order derivative of the function \(y = a^{mx}\)
    \({\left( {{a^{mx}}} \right)^{\left( n \right)}} = {m^n}{a^{mx}}{\ln ^n}a\)
  13. \(N\)th order derivative of the sine function
    \({\left( {\sin x} \right)^{\left( n \right)}} =\) \( \sin \left( {x + {\large\frac{{\pi n}}{2}}\normalsize} \right)\)
  14. \(N\)th order derivative of the cosine function
    \({\left( {\cos x} \right)^{\left( n \right)}} =\) \( \cos \left( {x + {\large\frac{{\pi n}}{2}}\normalsize} \right)\)