Calculus/Tables of Derivatives

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Contents

[edit] General Rules

\frac{d}{dx}(f + g)= \frac{df}{dx} + \frac{dg}{dx}

\frac{d}{dx}(cf)= c\frac{df}{dx}

\frac{d}{dx}(fg)= f\frac{dg}{dx} + g\frac{df}{dx}

\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{g\frac{df}{dx} - f\frac{dg}{dx}}{g^2}

[edit] Powers and Polynomials

  • \frac{d}{dx} (c) = 0
  • \frac{d}{dx}x=1
  • \frac{d}{dx}x^n=nx^{n-1}
  • \frac{d}{dx}\sqrt{x}=\frac{1}{2\sqrt x}
  • \frac{d}{dx}\frac{1}{x}=-\frac{1}{x^2}
  • \frac{d}{dx}(c_n x^n + c_{n-1} x^{n-1} + c_{n-2}x^{n-2} + \cdots +c_2x^2 + c_1 x + c_0) = n c_n x^{n-1} + (n-1) c_{n-1} x^{n-2} + (n-2) c_{n-2}x^{n-3} + \cdots + 2c_2x+ c_1

[edit] Trigonometric Functions

\frac{d}{dx} \sin (x)= \cos (x)

\frac{d}{dx} \cos (x)= -\sin (x)

\frac{d}{dx} \tan (x)= \sec^2 (x)

\frac{d}{dx} \cot (x)= -\csc^2 (x)

\frac{d}{dx} \sec (x)= \sec (x) \tan (x)

\frac{d}{dx} \csc (x) = -\csc (x) \cot (x)

[edit] Exponential and Logarithmic Functions

  • \frac{d}{dx} e^x =e^x
  • \frac{d}{dx} a^x =a^x \ln (a)\qquad\mbox{if }a>0
  • \frac{d}{dx} \ln (x)= \frac{1}{x}
  • \frac{d}{dx} \log_a (x)= \frac{1}{x\ln (a)}\qquad\mbox{if }a>0, a\neq 1
  • (f^g)' = \left(e^{g\ln f}\right)' = f^g\left(f'{g \over f} + g'\ln f\right),\qquad f > 0
  • (c^f)' = \left(e^{f\ln c}\right)' = f' c^f \ln c

[edit] Inverse Trigonometric Functions

  • \frac{d}{dx} \mbox{arcsin x} = \frac{1}{\sqrt{1-x^2}}
  • \frac{d}{dx} \mbox{arccos x} = -\frac{1}{\sqrt{1-x^2}}
  • \frac{d}{dx} \mbox{arctan x} = \frac{1}{1+x^2}
  • {d \over dx} \arcsec x = { 1 \over |x|\sqrt{x^2 - 1}}
  • {d \over dx} \arccot x = {-1 \over 1 + x^2}
  • {d \over dx} \arccsc x = {-1 \over |x|\sqrt{x^2 - 1}}

[edit] Hyperbolic and Inverse Hyperbolic Functions

{d \over dx} \sinh x = \cosh x
{d \over dx} \cosh x = \sinh x
{d \over dx} \tanh x = \mbox{sech}^2\,x
{d \over dx} \,\mbox{sech}\,x = -\tanh x\,\mbox{sech}\,x
{d \over dx} \,\mbox{coth}\,x = -\,\mbox{csch}^2\,x
{d \over dx} \,\mbox{csch}\,x = -\,\mbox{coth}\,x\,\mbox{csch}\,x
{d \over dx} \sinh^{-1} x = { 1 \over \sqrt{x^2 + 1}}
{d \over dx} \cosh^{-1} x = {-1 \over \sqrt{x^2 - 1}}
{d \over dx} \tanh^{-1} x = { 1 \over 1 - x^2}
{d \over dx} \mbox{sech}^{-1}\,x = { 1 \over x\sqrt{1 - x^2}}
{d \over dx} \mbox{coth}^{-1}\,x = {-1 \over 1 - x^2}
{d \over dx} \mbox{csch}^{-1}\,x = {-1 \over |x|\sqrt{1 + x^2}}

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