Table of derivatives

From Academic Kids

The primary operation in differential calculus is finding a derivative. This table lists derivatives of many functions. In the following, f and g are functions of x, and c is a constant. The set of real numbers is assumed. These formulas are sufficient to differentiate any elementary function.

Contents

Rules for differentiation of general functions

<math>\left({cf}\right)' = cf'<math>
<math>\left({f + g}\right)' = f' + g'<math>
<math>\left({f - g}\right)' = f' - g'<math>
<math>\left({fg}\right)' = f'g + fg'<math>
<math>\left({f \over g}\right)' = {f'g - fg' \over g^2}<math>
<math>(f^g)' = f^g\left(f'{g \over f} + g'\ln f\right),\qquad f > 0<math>
<math>(f \circ g)' = (f' \circ g)g'<math>

Derivatives of simple functions

<math>{d \over dx} c = 0<math>
<math>{d \over dx} x = 1<math>
<math>{d \over dx} |x| = {x \over |x|} = \sgn x,\qquad x \ne 0<math>
<math>{d \over dx} x^c = cx^{c-1}<math>
<math>{d \over dx} \sqrt{x} = {1 \over 2 \sqrt{x}}<math>
<math>{d \over dx} \left({1 \over x}\right) = -{1 \over x^2}<math>

Derivatives of exponential and logarithmic functions

<math>{d \over dx} c^x = {c^x \ln c},\qquad c > 0<math>
<math>{d \over dx} e^x = e^x<math>
<math>{d \over dx} \log_c x = {1 \over x \ln c},\qquad c > 0, c \ne 1<math>
<math>{d \over dx} \ln x = {1 \over x}<math>

Derivatives of trigonometric functions

<math>{d \over dx} \sin x = \cos x<math>
<math>{d \over dx} \cos x = -\sin x<math>
<math>{d \over dx} \tan x = \sec^2 x<math>
<math>{d \over dx} \sec x = \tan x \sec x<math>
<math>{d \over dx} \cot x = -\csc^2 x<math>
<math>{d \over dx} \csc x = -\cot x \csc x<math>
<math>{d \over dx} \arcsin x = { 1 \over \sqrt{1 - x^2}}<math>
<math>{d \over dx} \arccos x = {-1 \over \sqrt{1 - x^2}}<math>
<math>{d \over dx} \arctan x = { 1 \over 1 + x^2}<math>
<math>{d \over dx} \arcsec x = { 1 \over |x|\sqrt{x^2 - 1}}<math>
<math>{d \over dx} \arccot x = {-1 \over 1 + x^2}<math>
<math>{d \over dx} \arccsc x = {-1 \over |x|\sqrt{x^2 - 1}}<math>

Derivatives of hyperbolic functions

<math>{d \over dx} \sinh x = \cosh x<math>
<math>{d \over dx} \cosh x = \sinh x<math>
<math>{d \over dx} \tanh x = \mbox{sech}^2\,x<math>
<math>{d \over dx} \,\mbox{sech}\,x = -\tanh x\,\mbox{sech}\,x<math>
<math>{d \over dx} \,\mbox{coth}\,x = -\,\mbox{csch}^2\,x<math>
<math>{d \over dx} \,\mbox{csch}\,x = -\,\mbox{coth}\,x\,\mbox{csch}\,x<math>
<math>{d \over dx} \sinh^{-1} x = { 1 \over \sqrt{x^2 + 1}}<math>
<math>{d \over dx} \cosh^{-1} x = {-1 \over \sqrt{x^2 - 1}}<math>
<math>{d \over dx} \tanh^{-1} x = { 1 \over 1 - x^2}<math>
<math>{d \over dx} \mbox{sech}^{-1}\,x = { 1 \over x\sqrt{1 - x^2}}<math>
<math>{d \over dx} \mbox{coth}^{-1}\,x = {-1 \over 1 - x^2}<math>
<math>{d \over dx} \mbox{csch}^{-1}\,x = {-1 \over |x|\sqrt{1 + x^2}}<math>ro:Tabel de derivate
Navigation

Academic Kids Menu

  • Art and Cultures
    • Art (http://www.academickids.com/encyclopedia/index.php/Art)
    • Architecture (http://www.academickids.com/encyclopedia/index.php/Architecture)
    • Cultures (http://www.academickids.com/encyclopedia/index.php/Cultures)
    • Music (http://www.academickids.com/encyclopedia/index.php/Music)
    • Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
  • Biographies (http://www.academickids.com/encyclopedia/index.php/Biographies)
  • Clipart (http://www.academickids.com/encyclopedia/index.php/Clipart)
  • Geography (http://www.academickids.com/encyclopedia/index.php/Geography)
    • Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
    • Maps (http://www.academickids.com/encyclopedia/index.php/Maps)
    • Flags (http://www.academickids.com/encyclopedia/index.php/Flags)
    • Continents (http://www.academickids.com/encyclopedia/index.php/Continents)
  • History (http://www.academickids.com/encyclopedia/index.php/History)
    • Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
    • Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
    • Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
    • Prehistory (http://www.academickids.com/encyclopedia/index.php/Prehistory)
    • Renaissance (http://www.academickids.com/encyclopedia/index.php/Renaissance)
    • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
    • United States (http://www.academickids.com/encyclopedia/index.php/United_States)
    • Wars (http://www.academickids.com/encyclopedia/index.php/Wars)
    • World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
  • Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
  • Mathematics (http://www.academickids.com/encyclopedia/index.php/Mathematics)
  • Reference (http://www.academickids.com/encyclopedia/index.php/Reference)
  • Science (http://www.academickids.com/encyclopedia/index.php/Science)
    • Animals (http://www.academickids.com/encyclopedia/index.php/Animals)
    • Aviation (http://www.academickids.com/encyclopedia/index.php/Aviation)
    • Dinosaurs (http://www.academickids.com/encyclopedia/index.php/Dinosaurs)
    • Earth (http://www.academickids.com/encyclopedia/index.php/Earth)
    • Inventions (http://www.academickids.com/encyclopedia/index.php/Inventions)
    • Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
    • Plants (http://www.academickids.com/encyclopedia/index.php/Plants)
    • Scientists (http://www.academickids.com/encyclopedia/index.php/Scientists)
  • Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
    • Anthropology (http://www.academickids.com/encyclopedia/index.php/Anthropology)
    • Economics (http://www.academickids.com/encyclopedia/index.php/Economics)
    • Government (http://www.academickids.com/encyclopedia/index.php/Government)
    • Religion (http://www.academickids.com/encyclopedia/index.php/Religion)
    • Holidays (http://www.academickids.com/encyclopedia/index.php/Holidays)
  • Space and Astronomy
    • Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)
    • Planets (http://www.academickids.com/encyclopedia/index.php/Planets)
  • Sports (http://www.academickids.com/encyclopedia/index.php/Sports)
  • Timelines (http://www.academickids.com/encyclopedia/index.php/Timelines)
  • Weather (http://www.academickids.com/encyclopedia/index.php/Weather)
  • US States (http://www.academickids.com/encyclopedia/index.php/US_States)

Information

  • Home Page (http://academickids.com/encyclopedia/index.php)
  • Contact Us (http://www.academickids.com/encyclopedia/index.php/Contactus)

  • Clip Art (http://classroomclipart.com)
Toolbox
Personal tools