Table Of Derivatives

Rules for differentiation of general functions

$\backslash left(\{cf\}\backslash right)\text{'}\; =\; cf\text{'}$

$\backslash left(\{f\; +\; g\}\backslash right)\text{'}\; =\; f\text{'}\; +\; g\text{'}$

$\backslash left(\{f\; -\; g\}\backslash right)\text{'}\; =\; f\text{'}\; -\; g\text{'}$

$\backslash left(\{fg\}\backslash right)\text{'}\; =\; f\text{'}g\; +\; fg\text{'}$

$\backslash left(\{f\; \backslash over\; g\}\backslash right)\text{'}\; =\; \{f\text{'}g\; -\; fg\text{'}\; \backslash over\; g^2\}$

$(f^g)\text{'}\; =\; f^g\backslash left(f\text{'}\{g\; \backslash over\; f\}\; +\; g\text{'}\backslash ln\; f\backslash right),\backslash qquad\; f\; >\; 0$

$(f\; \backslash circ\; g)\text{'}\; =\; (f\text{'}\; \backslash circ\; g)g\text{'}$

Derivatives of simple functions

$\{d\; \backslash over\; dx\}\; c\; =\; 0$

$\{d\; \backslash over\; dx\}\; x\; =\; 1$

$\{d\; \backslash over\; dx\}\; |x|\; =\; \{x\; \backslash over\; |x|\}\; =\; \backslash sgn\; x,\backslash qquad\; x\; \backslash ne\; 0$

$\{d\; \backslash over\; dx\}\; x^c\; =\; cx^\{c-1\}$

$\{d\; \backslash over\; dx\}\; \backslash sqrt\{x\}\; =\; \{1\; \backslash over\; 2\; \backslash sqrt\{x\}\}$

$\{d\; \backslash over\; dx\}\; \backslash left(\{1\; \backslash over\; x\}\backslash right)\; =\; -\{1\; \backslash over\; x^2\}$

Derivatives of exponential and logarithmic functions

$\{d\; \backslash over\; dx\}\; c^x\; =\; \{c^x\; \backslash ln\; c\},\backslash qquad\; c\; >\; 0$

$\{d\; \backslash over\; dx\}\; e^x\; =\; e^x$

$\{d\; \backslash over\; dx\}\; \backslash log\_c\; x\; =\; \{1\; \backslash over\; x\; \backslash ln\; c\},\backslash qquad\; c\; >\; 0,\; c\; \backslash ne\; 1$

$\{d\; \backslash over\; dx\}\; \backslash ln\; x\; =\; \{1\; \backslash over\; x\}$

$\{d\; \backslash over\; dx\}\; \backslash left\backslash over\; (\backslash ln\; x)^n\}\; \backslash right=\; \backslash frac\{-n\}\{x\backslash ,(\backslash ln\; x)^\{n+1\}\}$

Derivatives of trigonometric functions

$\{d\; \backslash over\; dx\}\; \backslash sin\; x\; =\; \backslash cos\; x$

$\{d\; \backslash over\; dx\}\; \backslash cos\; x\; =\; -\backslash sin\; x$

$\{d\; \backslash over\; dx\}\; \backslash tan\; x\; =\; \backslash sec^2\; x$

$\{d\; \backslash over\; dx\}\; \backslash sec\; x\; =\; \backslash tan\; x\; \backslash sec\; x$

$\{d\; \backslash over\; dx\}\; \backslash cot\; x\; =\; -\backslash csc^2\; x$

$\{d\; \backslash over\; dx\}\; \backslash csc\; x\; =\; -\backslash cot\; x\; \backslash csc\; x$

$\{d\; \backslash over\; dx\}\; \backslash sin^\{-1\}\; x\; =\; \{\; 1\; \backslash over\; \backslash sqrt\{1\; -\; x^2\}\}$

$\{d\; \backslash over\; dx\}\; \backslash cos^\{-1\}\; x\; =\; \{-1\; \backslash over\; \backslash sqrt\{1\; -\; x^2\}\}$

$\{d\; \backslash over\; dx\}\; \backslash tan^\{-1\}\; x\; =\; \{\; 1\; \backslash over\; 1\; +\; x^2\}$

$\{d\; \backslash over\; dx\}\; \backslash sec^\{-1\}\; x\; =\; \{\; 1\; \backslash over\; |x|\backslash sqrt\{x^2\; -\; 1\}\}$

$\{d\; \backslash over\; dx\}\; \backslash cot^\{-1\}\; x\; =\; \{-1\; \backslash over\; 1\; +\; x^2\}$

$\{d\; \backslash over\; dx\}\; \backslash csc^\{-1\}\; x\; =\; \{-1\; \backslash over\; |x|\backslash sqrt\{x^2\; -\; 1\}\}$

Derivatives of hyperbolic functions

$\{d\; \backslash over\; dx\}\; \backslash sinh\; x\; =\; \backslash cosh\; x$

$\{d\; \backslash over\; dx\}\; \backslash cosh\; x\; =\; \backslash sinh\; x$

$\{d\; \backslash over\; dx\}\; \backslash tanh\; x\; =\; \backslash mbox\{sech\}^2\backslash ,x$

$\{d\; \backslash over\; dx\}\; \backslash ,\backslash mbox\{sech\}\backslash ,x\; =\; -\backslash tanh\; x\backslash ,\backslash mbox\{sech\}\backslash ,x$

$\{d\; \backslash over\; dx\}\; \backslash ,\backslash mbox\{coth\}\backslash ,x\; =\; -\backslash ,\backslash mbox\{csch\}^2\backslash ,x$

$\{d\; \backslash over\; dx\}\; \backslash ,\backslash mbox\{csch\}\backslash ,x\; =\; -\backslash ,\backslash mbox\{coth\}\backslash ,x\backslash ,\backslash mbox\{csch\}\backslash ,x$

$\{d\; \backslash over\; dx\}\; \backslash sinh^\{-1\}\; x\; =\; \{\; 1\; \backslash over\; \backslash sqrt\{x^2\; +\; 1\}\}$

$\{d\; \backslash over\; dx\}\; \backslash cosh^\{-1\}\; x\; =\; \{-1\; \backslash over\; \backslash sqrt\{x^2\; -\; 1\}\}$

$\{d\; \backslash over\; dx\}\; \backslash tanh^\{-1\}\; x\; =\; \{\; 1\; \backslash over\; 1\; -\; x^2\}$

$\{d\; \backslash over\; dx\}\; \backslash mbox\{sech\}^\{-1\}\backslash ,x\; =\; \{\; 1\; \backslash over\; x\backslash sqrt\{1\; -\; x^2\}\}$

$\{d\; \backslash over\; dx\}\; \backslash mbox\{coth\}^\{-1\}\backslash ,x\; =\; \{-1\; \backslash over\; 1\; -\; x^2\}$

$\{d\; \backslash over\; dx\}\; \backslash mbox\{csch\}^\{-1\}\backslash ,x\; =\; \{-1\; \backslash over\; |x|\backslash sqrt\{1\; +\; x^2\}\}$

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