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