Calculus/Tables of Derivatives

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General Rules[edit]

\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}

 [f(g(x))]' = f'(g(x)) g'(x)

Powers and Polynomials[edit]

  • \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}

Trigonometric Functions[edit]

\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)

Exponential and Logarithmic Functions[edit]

  • \frac{d}{dx} e^x =e^x
  • \frac{d}{dx} a^x =a^x \ln (a)\qquad\text{if }a>0
  • \frac{d}{dx} \ln (x)= \frac{1}{x}
  • \frac{d}{dx} \log_a (x)= \frac{1}{x\ln (a)}\qquad\text{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

Inverse Trigonometric Functions[edit]

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

Hyperbolic and Inverse Hyperbolic Functions[edit]

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