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