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differentiation rules

Here follow the rules for differentiating common functions of a single real variable that every student of calculus should know. In each case the variable with respect to which the derivative is being taken is \(x\), both \(u\) and \(v\) are functions of \(x\),and \(c\) is a constant. Note that \(u^{\prime}\) should be understood as \(\displaystyle\frac{du}{dx}\).

\[ \begin{eqnarray*} \frac{d}{dx}\left( c \right) & = & 0 \\ & & \\ \frac{d}{dx}\left( x \right) & = & 1 \\ & & \\ \frac{d}{dx}\left( cu \right) & = & cu^{\prime} \\ & & \\ \frac{d}{dx}\left( u \pm v \right) & = & u^{\prime} + v^{\prime} \\ & & \\ \frac{d}{dx}\left( uv \right) & = & u^{\prime}v + uv^{\prime} \\ & & \\ \frac{d}{dx}\left( \frac{u}{v} \right) & = & \frac{u^{\prime}v - uv^{\prime}}{v^2} \\ & & \\ \frac{d}{dx}\left( u^n \right) & = & nu^{n-1}u^{\prime} \\ & & \\ \frac{d}{dx}\left( \ln u \right) & = & \frac{u^{\prime}}{u} \\ & & \\ \frac{d}{dx}\left( \sin u \right) & = & (\sin u)u^{\prime} \\ & & \\ \frac{d}{dx}\left( \cos u \right) & = & (-\sin u)u^{\prime} \\ & & \\ \frac{d}{dx}\left( \tan u \right) & = & \left(\sec^2 u \right) u^{\prime} \\ & & \\ \frac{d}{dx}\left( \cot u \right) & = & -\left( \csc^2 u\right) u^{\prime} \\ & & \\ \frac{d}{dx}\left( \sec u \right) & = & (\sec u \tan u)u^{\prime} \\ & & \\ \frac{d}{dx}\left( \csc u \right) & = & -(\csc u \cot u)u^{\prime} \\ & & \\ \frac{d}{dx}\left( \arcsin u \right) & = & \frac{u^{\prime}}{\sqrt{1-u^2}} \\ & & \\ \frac{d}{dx}\left( \arccos u \right) & = & \frac{-u^{\prime}}{\sqrt{1-u^2}} \\ & & \\ \frac{d}{dx}\left( \arctan u \right) & = & \frac{u^{\prime}}{1+u^2} \\ & & \\ \frac{d}{dx}\left( \mbox{arccot } u \right) & = & \frac{-u^{\prime}}{1+u^2} \\ & & \\ \frac{d}{dx}\left( \mbox{arcsec } u \right) & = & \frac{u^{\prime}}{|u|\sqrt{u^2-1}} \\ & & \\ \frac{d}{dx}\left( \mbox{arccsc } u \right) & = & \frac{-u^{\prime}}{|u|\sqrt{u^2-1}} \\ & & \\ \end{eqnarray*} \]

Citation Info

  • [MLA] “differentiation rules.” Platonic Realms Interactive Mathematics Encyclopedia. Platonic Realms, 28 Feb 2013. Web. 28 Feb 2013. <http://platonicrealms.com/>
  • [APA] differentiation rules (28 Feb 2013). Retrieved 28 Feb 2013 from the Platonic Realms Interactive Mathematics Encyclopedia: http://platonicrealms.com/encyclopedia/differentiation-rules/

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