Appendix D: Derivative Facts
- Page ID
- 216422
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Basic Patterns
\(\mbox{D}(k) = 0\) (\(k\) represents a constant)
\(\mbox{D}(k\cdot f) = k\cdot\mbox{D}(f)\)
\(\mbox{D}(f + g) = \mbox{D}(f) + \mbox{D}(g)\)
\(\mbox{D}(f - g) = \mbox{D}(f) - \mbox{D}(g)\)
\(\mbox{D}(f\cdot g) = f\cdot\mbox{D}(g) + g\cdot\mbox{D}(f)\) (Product Rule)
\(\displaystyle \mbox{D}\left(\frac{f}{g}\right) = \frac{g\cdot\mbox{D}(f) - f\cdot\mbox{D}(g)}{g^2}\) (Quotient Rule)
Power Rules
\(\mbox{D}(x^p) = p\cdot x^{p-1}\)
\(\mbox{D}\left(f^n\right) = n\cdot f^{n-1}\cdot\mbox{D}(f)\)
Chain Rule
\(\mbox{D}\left(f(g(x))\right) = f'\left(g(x)\right)\cdot g'(x)\)
\(\displaystyle \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)
Exponential and Logarithmic Functions
\(\mbox{D}\left(e^u\right) = e^u\)/p>
\(\mbox{D}\left(a^u\right) = a^u \cdot\ln(a)\)
\(\displaystyle \mbox{D}\left(\ln(\left|u\right|)\right) = \frac{1}{u}\)
\(\displaystyle \mbox{D}\left(\log_a(\left| u \right|)\right) = \frac{1}{u\cdot\ln(a)}\)
\(\displaystyle \mbox{D}\left(\ln(f(x))\right) = \frac{f'(x)}{f(x)}\)
Trigonometric Functions
\(\mbox{D}\left(\sin(u))\right) = \cos(u))\)
\(\mbox{D}(\tan(u )) = \sec^2(u)\)
\(\mbox{D}(\sec(u)) = \sec(u)\cdot \tan(u)\)
\(\mbox{D}(\cos(u)) = -\sin(u)\)
\(\mbox{D}(\cot(u)) = -\csc^2(u)\)
\(\mbox{D}(\csc(u)) = -\csc(u)\cdot\cot(u)\)
Inverse Trigonometric Functions
\(\displaystyle \mbox{D}(\arcsin( u ) ) = \frac{1}{\sqrt{1 - u^2}}\)
\(\displaystyle \mbox{D}(\arctan( u ) ) = \frac{1}{1 + u^2}\)
\(\displaystyle \mbox{D}(\mbox{arcsec}( u ) ) = \frac{1}{\left|u\right| \sqrt{u^2-1}}\)
\(\displaystyle \mbox{D}(\arccos( u ) ) = \frac{-1}{\sqrt{1-u^2}}\)
\(\displaystyle \mbox{D}(\mbox{arccot}( u ) ) = \frac{-1}{1 + u^2}\)
\(\displaystyle \mbox{D}(\mbox{arccsc}( u ) ) = \frac{-1}{\left|u\right| \sqrt{u^2-1}}\)
Hyperbolic Functions
\(\mbox{D}\left(\sinh(u))\right) = \cosh(u))\)
\(\mbox{D}(\cosh(u)) = \sinh(u)\)
\(\mbox{D}(\tanh(u )) = \mbox{sech}^2(u)\)
\(\mbox{D}(\coth(u)) = -\mbox{csch}^2(u)\)
\(\mbox{D}(\mbox{sech}(u)) = -\mbox{sech}(u)\cdot \tanh(u)\)
\(\mbox{D}(\mbox{csch}(u)) = -\mbox{csch}(u)\cdot\coth(u)\)
Inverse Hyperbolic Functions
\(\displaystyle \mbox{D}(\mbox{argsinh}( u ) ) = \frac{1}{\sqrt{1 + u^2}}\)
\(\displaystyle \mbox{D}(\mbox{argcosh}( u ) ) = \frac{1}{\sqrt{u^2-1}}\) (for \(u> 1\))
\(\displaystyle \mbox{D}(\mbox{argtanh}( u ) ) = \frac{1}{1 - u^2}\) (for \(\left|u\right| < 1\))
\(\displaystyle \mbox{D}(\mbox{argcoth}( u ) ) = \frac{1}{1 - u^2}\) (for \(\left|u\right| > 1\))
\(\displaystyle \mbox{D}(\mbox{argsech}( u ) ) = \frac{-1}{\left|u\right| \sqrt{1-u^2}}\) (for \(0 < u < 1\))
\(\displaystyle \mbox{D}(\mbox{argcsch}( u ) ) = \frac{-1}{\left|u\right| \sqrt{u^2+1}}\) (for \(u \neq 0\))