# Results of Higher Derivatives

1) ${y_n} = \frac{{{d^n}}}{{d{x^n}}}{(ax + b)^m} = \frac{{m!}}{{(m – n)!}}{a^n}{(ax + b)^{m – n}}$

2) ${y_n} = \frac{{{d^n}}}{{d{x^n}}}\frac{1}{{(ax + b)}} = \frac{{{{( – 1)}^n}n!{a^n}}}{{{{\left( {ax + b} \right)}^{n + 1}}}}$

3) ${y_n} = \frac{{{d^n}}}{{d{x^n}}}\ln \left( {ax + b} \right) = \frac{{{{\left( { – 1} \right)}^n}\left( {n – 1} \right)!{a^n}}}{{{{\left( {ax + b} \right)}^n}}}$

4) ${y_n} = \frac{{{d^n}}}{{d{x^n}}}Sin\left( {ax + b} \right) = {a^n}Sin\left( {ax + b + n\frac{\pi }{2}} \right)$

5) ${y_n} = \frac{{{d^n}}}{{d{x^n}}}Cos\left( {ax + b} \right) = {a^n}Cos\left( {ax + b + n\frac{\pi }{2}} \right)$

6) ${y_n} = \frac{{{d^n}}}{{d{x^n}}}{e^{ax}} = {a^n}{e^{ax}}$

7) ${y_n} = \frac{{{d^n}}}{{d{x^n}}}{e^{ax}}Sin\left( {ax + c} \right) = {\left( {{a^2} + {b^2}} \right)^{\frac{n}{2}}}{e^{ax}}Sin\left( {bx + c + nTa{n^{ – 1}}\frac{a}{b}} \right)$

8) ${y_n} = \frac{{{d^n}}}{{d{x^n}}}{e^{ax}}Cos\left( {ax + c} \right) = {\left( {{a^2} + {b^2}} \right)^{\frac{n}{2}}}{e^{ax}}Cos\left( {bx + c + nTa{n^{ – 1}}\frac{a}{b}} \right)$

9)  If $y = {\left( {ax + b} \right)^n}$, then ${y_{n + r}} = 0 for  r > 0$

Leibniz’s Theorem

${\left( {f \cdot g} \right)_n} = {f_n} \cdot g + n{f_{n – 1}} \cdot {g_1} + \frac{{n\left( {n – 1} \right)}}{{2!}}{f_{n – 2}} \cdot {g_2} + \cdots + f \cdot {g_n}$

Taylor’s Theorem

$f\left( {x + h} \right) = f\left( x \right) + h{f_1}\left( x \right) + \frac{{{h^2}}}{{2!}}{f_2}\left( x \right) + \frac{{{h^3}}}{{3!}}{f_3}\left( x \right) + \cdots + \frac{{{h^n}}}{{n!}}{f_n}\left( x \right) + \cdots$

Meclaurin’s Series

$f\left( x \right) = f\left( 0 \right) + x{f_1}\left( 0 \right) + \frac{{{x^2}}}{{2!}}{f_2}\left( 0 \right) + \frac{{{x^3}}}{{3!}}{f_3}\left( 0 \right) + \cdots + \frac{{{x^n}}}{{n!}}{f_n}\left( 0 \right) + \cdots$