Subject: Calculus

Table Of Derivatives

Calculus.TableOfDerivatives History

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July 03, 2011 by matthew_suan -
July 03, 2011 by matthew_suan -
Added lines 112-113:

'''NEXT TABLE''': [[Table of integrals]]
December 03, 2010 by matthew_suan -
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!!Derivative of ordinary functions

(
:table border=3 cellpadding=3 cellspacing=0 align=center:)
to:
(:table border=1 cellpadding=5 cellspacing=0:)
(:cellnr
bgcolor=#d4d7ba colspan=14 align=center:) '+'''The Page at a Glance'''+'
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*[[#t1 | Derivative of ordinary functions]]
*[[#t2 | Derivative of ordinary exponential and logarithmic functions]]
*[[#t3 | Derivative involving trigonometric functions]]
*[[#t4 | Derivative involving inverse trigonometric functions]]
*[[#t5 | Derivative involving hyperbolic trigonometric functions]]
(:tableend:)

[[#t1]]
!!Derivative of ordinary functions

(:table border=3 cellpadding=3 cellspacing=0 align=center:)
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[[#t2]]
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[[#t3]]
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[[#t4]]
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to:
[[#t5]]
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%center%Inverse trigonometric functions
to:
%center%Hyperbolic trigonometric functions
December 02, 2010 by matthew_suan -
Changed lines 85-95 from:
1.{$D_{x}\sinh{u} = \cosh{u}D_{x}u$}

2.{$D_{x}\cosh{u} = \sinh{u}D_{x}u$}

3.{$D_{x}\tanh{u} = sech^{2}{u}D_{x}u$}

4.{$D_{x}\coth{u} = -csch^{2}{u}D_{x}u$}

5.{$D_{x}sech{u} = -sech{u}tanh{u}D_{x}u$}

6.{$D_{x}csch{u} = -csch{u}coth{u}D_{x}u$}
to:
1.{$D_{x}\sinh{(u)} = \cosh{(u)}D_{x}u$}

2.{$D_{x}\cosh{(u)} = \sinh{(u)}D_{x}u$}

3.{$D_{x}\tanh{(u)} = sech^{2}{(u)}D_{x}u$}

4.{$D_{x}\coth{(u)} = -csch^{2}{(u)}D_{x}u$}

5.{$D_{x}sech{(u)} = -sech{(u)}tanh{(u)}D_{x}u$}

6.{$D_{x}csch{(u)} = -csch{(u)}\coth{(u)}D_{x}u$}
December 02, 2010 by matthew_suan -
Changed lines 89-95 from:
3.{$D_{x}\tanh{u} = \sech^{2}{u}D_{x}u$}

4.{$D_{x}\coth{u} = -\csch^{2}{u}D_{x}u$}

5.{$D_{x}\sech{u} = -\sech{u}\tanh{u}D_{x}u$}

6.{$D_{x}\csch{u} = -\csch{u}\coth{u}D_{x}u$}
to:
3.{$D_{x}\tanh{u} = sech^{2}{u}D_{x}u$}

4.{$D_{x}\coth{u} = -csch^{2}{u}D_{x}u$}

5.{$D_{x}sech{u} = -sech{u}tanh{u}D_{x}u$}

6.{$D_{x}csch{u} = -csch{u}coth{u}D_{x}u$}
December 02, 2010 by matthew_suan -
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Deleted lines 31-32:
Deleted line 52:
Changed lines 85-95 from:
1.{$D_{x}\sin^{-1}{u}$} = {$\frac{1}{\sqrt{1-u^{2}}}D_{x}u$}

2.
{$D_{x}\cos^{-1}{u}$} = -{$\frac{1}{\sqrt{1-u^{2}}}D_{x}u$}

3.{$D_{x}\tan^{-1}{u}$} = {$\frac{1}{1+u^{2}}D_{x}u$}

4.{$D_{x}\cot^{-1}{u}$} = -{$\frac{1}{1+u^{2}}D_{x}u$}

5.
{$D_{x}\sec^{-1}{u}$} = {$\frac{1}{u\sqrt{u^{2}-1}}D_{x}u$}

6.{$D_{x}\csc^{-1}{u}$} = -{$\frac{1}{u\sqrt{u^{2}-1}
}D_{x}u$}
to:
1.{$D_{x}\sinh{u} = \cosh{u}D_{x}u$}

2.{$
D_{x}\cosh{u} = \sinh{u}D_{x}u$}

3.
{$D_{x}\tanh{u} = \sech^{2}{u}D_{x}u$}

4.{$D_{x}\coth{u} = -\csch^{2}{u}D_{x}u$}

5.{$D_{x}\sech{u} = -\sech{u}\tanh{u}D_{x}u$}

6.{$
D_{x}\csch{u} = -\csch{u}\coth{u}D_{x}u$}
December 02, 2010 by matthew_suan -
Added lines 61-82:

(:table border=3 cellpadding=3 cellspacing=0 align=center:)
(:cellnr:)
%center%Inverse trigonometric functions
(:cellnr:)

1.{$D_{x}\sin^{-1}{u}$} = {$\frac{1}{\sqrt{1-u^{2}}}D_{x}u$}

2.{$D_{x}\cos^{-1}{u}$} = -{$\frac{1}{\sqrt{1-u^{2}}}D_{x}u$}

3.{$D_{x}\tan^{-1}{u}$} = {$\frac{1}{1+u^{2}}D_{x}u$}

4.{$D_{x}\cot^{-1}{u}$} = -{$\frac{1}{1+u^{2}}D_{x}u$}

5.{$D_{x}\sec^{-1}{u}$} = {$\frac{1}{u\sqrt{u^{2}-1}}D_{x}u$}

6.{$D_{x}\csc^{-1}{u}$} = -{$\frac{1}{u\sqrt{u^{2}-1}}D_{x}u$}
(:tableend:)
[[<<]]


!!Derivative involving hyperbolic trigonometric functions
December 02, 2010 by matthew_suan -
Changed lines 75-77 from:
5.{$D_{x}\sec^{-1}{u}$} = \sec{u}\tan{u}D_{x}u$}

6.
{$D_{x}\csc^{-1}{u}$} = -\csc{u}\cot{u}D_{x}u$}
to:
5.{$D_{x}\sec^{-1}{u}$} = {$\frac{1}{u\sqrt{u^{2}-1}}D_{x}u$}

6.
{$D_{x}\csc^{-1}{u}$} = -{$\frac{1}{u\sqrt{u^{2}-1}}D_{x}u$}
December 02, 2010 by matthew_suan -
Changed line 73 from:
4.{$D_{x}\cot^{-1}{u}$} = -\csc^{2}{u}D_{x}u$}
to:
4.{$D_{x}\cot^{-1}{u}$} = -{$\frac{1}{1+u^{2}}D_{x}u$}
December 02, 2010 by matthew_suan -
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to:
[[<<]]
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to:
[[<<]]
December 02, 2010 by matthew_suan -
Deleted lines 79-94:
!!Limits involving logarithmic and exponential functions

(:table border=3 cellpadding=3 cellspacing=0 align=center width=50%:)
(:cellnr:)
%center%Simple Functions logarithmic and exponential functions
(:cellnr:)

||1.{$\lim_{x \to 0^+}\log_{b}{x}=-\infty$} || ||

||2.{$\lim_{x \to \infty}\log_{b}{x}=\infty$} || ||

||3.{$\lim_{x \to -\infty }k^x=0$} ||; {$k$} is a constant ||

||4.{$\lim_{x \to \infty }k^x=\infty$} ||; {$k$} is a constant ||

(:tableend:)
December 02, 2010 by matthew_suan -
Changed lines 45-56 from:
1.{$D_{x}\sin{u}=\cos{u}D_{x}u$}

2.{$D_{x}\cos{u}=-\sin{u}D_{x}u$}

3.{$D_{x}\tan{u}=\sec^{2}{u}D_{x}u$}

4.{$D_{x}\cot{u}=-\csc^{2}{u}D_{x}u$}

5.{$D_{x}\sec{u}=\sec{u}\tan{u}D_{x}u$}

6.{$D_{x}\csc{u}=-\csc{u}\cot{u}D_{x}u$}
to:
1.{$D_{x}\sin{u} = \cos{u}D_{x}u$}

2.{$D_{x}\cos{u} = -\sin{u}D_{x}u$}

3.{$D_{x}\tan{u} = \sec^{2}{u}D_{x}u$}

4.{$D_{x}\cot{u} = -\csc^{2}{u}D_{x}u$}

5.{$D_{x}\sec{u} = \sec{u}\tan{u}D_{x}u$}

6.{$D_{x}\csc{u} = -\csc{u}\cot{u}D_{x}u$}
Changed lines 67-77 from:
1.{$D_{x}\sin^{-1}{u}$}={$\frac{1}{\sqrt{1-u^{2}}}D_{x}u$}

2.{$D_{x}\cos^{-1}{u}$}=-{$\frac{1}{\sqrt{1-u^{2}}}D_{x}u$}

3.{$D_{x}\tan^{-1}{u}$}={$\frac{1}{1+u^{2}}D_{x}u$}

4.{$D_{x}\cot^{-1}{u}$}=-\csc^{2}{u}D_{x}u$}

5.{$D_{x}\sec^{-1}{u}$}=\sec{u}\tan{u}D_{x}u$}

6.{$D_{x}\csc^{-1}{u}$}=-\csc{u}\cot{u}D_{x}u$}
to:
1.{$D_{x}\sin^{-1}{u}$} = {$\frac{1}{\sqrt{1-u^{2}}}D_{x}u$}

2.{$D_{x}\cos^{-1}{u}$} = -{$\frac{1}{\sqrt{1-u^{2}}}D_{x}u$}

3.{$D_{x}\tan^{-1}{u}$} = {$\frac{1}{1+u^{2}}D_{x}u$}

4.{$D_{x}\cot^{-1}{u}$} = -\csc^{2}{u}D_{x}u$}

5.{$D_{x}\sec^{-1}{u}$} = \sec{u}\tan{u}D_{x}u$}

6.{$D_{x}\csc^{-1}{u}$} = -\csc{u}\cot{u}D_{x}u$}
December 02, 2010 by matthew_suan -
Changed line 71 from:
3.{$D_{x}\tan^{-1}{u}$}=\sec^{2}{u}D_{x}u$}
to:
3.{$D_{x}\tan^{-1}{u}$}={$\frac{1}{1+u^{2}}D_{x}u$}
December 02, 2010 by matthew_suan -
Changed line 64 from:
%center%If {$\lim_{x \to k}f(x)=L_1$} and {$\lim_{x \to k}g(x)=L_2$}, then
to:
%center%Inverse trigonometric functions
Changed line 69 from:
2.{$D_{x}\cos^{-1}{u}$}=-\sin{u}D_{x}u$}
to:
2.{$D_{x}\cos^{-1}{u}$}=-{$\frac{1}{\sqrt{1-u^{2}}}D_{x}u$}
December 02, 2010 by matthew_suan -
Changed line 62 from:
(:table border=3 cellpadding=3 cellspacing=0 align=center width=50%:)
to:
(:table border=3 cellpadding=3 cellspacing=0 align=center:)
Changed lines 67-77 from:
1.{$D_{x}\sin^{-1}{u}=\cos{u}D_{x}u$}

2.{$
D_{x}\cos^{-1}{u}=-\sin{u}D_{x}u$}

3.{
$D_{x}\tan^{-1}{u}=\sec^{2}{u}D_{x}u$}

4.{
$D_{x}\cot^{-1}{u}=-\csc^{2}{u}D_{x}u$}

5.{
$D_{x}\sec^{-1}{u}=\sec{u}\tan{u}D_{x}u$}

6.{
$D_{x}\csc^{-1}{u}=-\csc{u}\cot{u}D_{x}u$}
to:
1.{$D_{x}\sin^{-1}{u}$}={$\frac{1}{\sqrt{1-u^{2}}}D_{x}u$}

2.
{$D_{x}\cos^{-1}{u}$}=-\sin{u}D_{x}u$}

3.
{$D_{x}\tan^{-1}{u}$}=\sec^{2}{u}D_{x}u$}

4.{$
D_{x}\cot^{-1}{u}$}=-\csc^{2}{u}D_{x}u$}

5.
{$D_{x}\sec^{-1}{u}$}=\sec{u}\tan{u}D_{x}u$}

6.{$D_{x}\csc^{-1}{u}$
}=-\csc{u}\cot{u}D_{x}u$}
December 02, 2010 by matthew_suan -
Changed lines 60-61 from:
!!Common limits for general functions
to:
!!Derivative involving inverse trigonometric functions
Changed lines 67-76 from:
||1.{$\lim_{x \to k}[f(x)\pm g(x)]=L_1\pm L_{2}$} || ||

||
2.{$\lim_{x \to c}[f(x)\cdot g(x)]=L_1\cdot L_{2}$} || ||

||
3.{$\lim_{x \to k}\frac{f(x)}{g(x)}$}={$L_1\over L_{2}$} ||; {$L\neq 0$} ||

||4.
{$\lim_{x \to k}f(x)^{n}$}={$L_{1}^{n}$} ||; {$n > 0$} ||

||5.
{$\lim_{x \to k}f(x)^{\frac{1}{n}}$}={$L_{1}^{\frac{1}{n}}$} ||; {$n > 0$}, {$n$} is even and {$L_{1}$} > 0 ||
to:
1.{$D_{x}\sin^{-1}{u}=\cos{u}D_{x}u$}

2.{$D_{x}\cos^{-1}{u}=-\sin{u}D_{x}u$}

3.{$D_{x}\tan^{-1}{u}=\sec^{2}{u}D_{x}u$}

4.
{$D_{x}\cot^{-1}{u}=-\csc^{2}{u}D_{x}u$}

5.
{$D_{x}\sec^{-1}{u}=\sec{u}\tan{u}D_{x}u$}

6.
{$D_{x}\csc^{-1}{u}=-\csc{u}\cot{u}D_{x}u$}
December 02, 2010 by matthew_suan -
Changed line 42 from:
%center%Simple Trigonometric Functions
to:
%center%6 Trigonometric Functions
Changed line 55 from:
to:
6.{$D_{x}\csc{u}=-\csc{u}\cot{u}D_{x}u$}
December 02, 2010 by matthew_suan -
Changed lines 53-55 from:
||5.{$\lim_{x \to k}\frac{1-\cos{x}}{x^2}=\frac{1}{2}$} || ||
to:
5.{$D_{x}\sec{u}=\sec{u}\tan{u}D_{x}u$}
December 02, 2010 by matthew_suan -
Changed lines 19-20 from:
to:
[[<<]]
Changed lines 36-40 from:

!!Limits involving trigonometric functions

(:table border=3 cellpadding=3 cellspacing=0 align=center width=50%:)
to:
[[<<]]

!!Derivative involving trigonometric functions

(:table border=3 cellpadding=3 cellspacing=0 align=center:)
Changed lines 45-51 from:
||1.{$\lim_{x \to k}\sin{x}=\sin{k}$} || ||

||
2.{$\lim_{x \to k}\cos{x}=\cos{k}$} || ||

||
3.{$\lim_{x \to k}\frac{\sin{x}}{x}=1$} || ||

||
4.{$\lim_{x \to k}\frac{1-\cos{x}}{x}=0$} || ||
to:
1.{$D_{x}\sin{u}=\cos{u}D_{x}u$}

2.{$D_{x}\cos{u}=-\sin{u}D_{x}u$}

3.{$D_{x}\tan{u}=\sec^{2}{u}D_{x}u$}

4.{$D_{x}\cot{u}=-\csc^{2}{u}D_{x}u$}
December 02, 2010 by matthew_suan -
Changed line 30 from:
||2. {$D_{x}a^{u} = {$a^{u} \ln{u} D_{x}u$} ||
to:
||2. {$D_{x}a^{u} = a^{u} \ln{u} D_{x}u$} ||
December 02, 2010 by matthew_suan -
Changed lines 32-34 from:
||3. {$D_{x}(u\cdot v)$} = {$vD_{x}u+uD_{x}v$} ||

||4. {$D_{x}(\frac{u}{v})$} = {$vD_{x}u-uD_{x}v\over v^2$} ||
to:
||3. {$D_{x}(\ln{u})$} = {$\frac{1}{u}D_{x}u$} ||
December 02, 2010 by matthew_suan -
Changed lines 28-30 from:
||1. {$D_{x}e^{u} = {$e^{u}D_{x}u$} ||

||2. {$D_{x}e^{u} = {$e^{u}D_{x}u$} ||
to:
||1. {$D_{x}e^{u} = e^{u}D_{x}u$} ||

||2. {$D_{x}a^{u} = {$a^{u} \ln{u} D_{x}u$} ||
December 02, 2010 by matthew_suan -
Deleted lines 0-1:
[[<<]]
Changed lines 10-17 from:
||1.{$D_{x}(u^{r})$} = {$ru^{n-1}\cdot D_{x}u$} ||

||2.{$D_{x}(u+v)$} = {$D_{x}u+D_{x}v$} ||

||3.{$\lim_{x \to k}ax + b=ak + b$} ||; {$a$} and {$b$} are constants ||

||4.
{$\lim_{x \to k}x^{r}$} = {$k^r$} ||; {$r > 0$} and an integer ||
to:
||1. {$D_{x}(u^{r})$} = {$ru^{n-1}\cdot D_{x}u$} ||

||2. {$D_{x}(u+v)$} = {$D_{x}u+D_{x}v$} ||

||3. {$D_{x}(u\cdot v)$} = {$vD_{x}u+uD_{x}v$} ||

||4.
{$D_{x}(\frac{u}{v})$} = {$vD_{x}u-uD_{x}v\over v^2$} ||

(:tableend:)


!!Derivative
of ordinary exponential and logarithmic functions

(:table
border=3 cellpadding=3 cellspacing=0 align=center:)
(:cellnr:)
%center%Simple Logarithmic and Exponential Functions
(:cellnr:)

||1. {$D_{x}e^{u} = {$e^{u}D_{x}u$} ||

||2. {$D_{x}e^{u} = {$e^{u}D_{x}u$} ||

||3. {$D_{x}(u\cdot v)$} = {$vD_{x}u+uD_{x}v$} ||

||4. {$D_{x}(\frac{u}{v})$} = {$vD_{x}u-uD_{x}v\over v^2$} ||
December 02, 2010 by matthew_suan -
Added lines 1-77:
[[<<]]

The following are the most common and simplest forms of derivative and its answers.

!!Derivative of ordinary functions

(:table border=3 cellpadding=3 cellspacing=0 align=center:)
(:cellnr:)
%center%Simple Functions
(:cellnr:)

||1.{$D_{x}(u^{r})$} = {$ru^{n-1}\cdot D_{x}u$} ||

||2.{$D_{x}(u+v)$} = {$D_{x}u+D_{x}v$} ||

||3.{$\lim_{x \to k}ax + b=ak + b$} ||; {$a$} and {$b$} are constants ||

||4.{$\lim_{x \to k}x^{r}$} = {$k^r$} ||; {$r > 0$} and an integer ||

(:tableend:)


!!Limits involving trigonometric functions

(:table border=3 cellpadding=3 cellspacing=0 align=center width=50%:)
(:cellnr:)
%center%Simple Trigonometric Functions
(:cellnr:)

||1.{$\lim_{x \to k}\sin{x}=\sin{k}$} || ||

||2.{$\lim_{x \to k}\cos{x}=\cos{k}$} || ||

||3.{$\lim_{x \to k}\frac{\sin{x}}{x}=1$} || ||

||4.{$\lim_{x \to k}\frac{1-\cos{x}}{x}=0$} || ||

||5.{$\lim_{x \to k}\frac{1-\cos{x}}{x^2}=\frac{1}{2}$} || ||

(:tableend:)


!!Common limits for general functions

(:table border=3 cellpadding=3 cellspacing=0 align=center width=50%:)
(:cellnr:)
%center%If {$\lim_{x \to k}f(x)=L_1$} and {$\lim_{x \to k}g(x)=L_2$}, then
(:cellnr:)

||1.{$\lim_{x \to k}[f(x)\pm g(x)]=L_1\pm L_{2}$} || ||

||2.{$\lim_{x \to c}[f(x)\cdot g(x)]=L_1\cdot L_{2}$} || ||

||3.{$\lim_{x \to k}\frac{f(x)}{g(x)}$}={$L_1\over L_{2}$} ||; {$L\neq 0$} ||

||4.{$\lim_{x \to k}f(x)^{n}$}={$L_{1}^{n}$} ||; {$n > 0$} ||

||5.{$\lim_{x \to k}f(x)^{\frac{1}{n}}$}={$L_{1}^{\frac{1}{n}}$} ||; {$n > 0$}, {$n$} is even and {$L_{1}$} > 0 ||

(:tableend:)

!!Limits involving logarithmic and exponential functions

(:table border=3 cellpadding=3 cellspacing=0 align=center width=50%:)
(:cellnr:)
%center%Simple Functions logarithmic and exponential functions
(:cellnr:)

||1.{$\lim_{x \to 0^+}\log_{b}{x}=-\infty$} || ||

||2.{$\lim_{x \to \infty}\log_{b}{x}=\infty$} || ||

||3.{$\lim_{x \to -\infty }k^x=0$} ||; {$k$} is a constant ||

||4.{$\lim_{x \to \infty }k^x=\infty$} ||; {$k$} is a constant ||

(:tableend:)