Appendix A: Table of Derivatives
- Page ID
- 14735
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)General Formulas
1. \(\quad \dfrac{d}{dx}\left(c\right)=0\)
2. \(\quad \dfrac{d}{dx}\left(f(x)+g(x)\right)=f′(x)+g′(x)\)
3. \(\quad \dfrac{d}{dx}\left(f(x)g(x)\right)=f′(x)g(x)+f(x)g′(x)\)
4. \(\quad \dfrac{d}{dx}\left(x^n\right)=nx^{n−1},\quad \text{for real numbers }n\)
5. \(\quad \dfrac{d}{dx}\left(cf(x)\right)=cf′(x)\)
6. \(\quad \dfrac{d}{dx}\left(f(x)−g(x)\right)=f′(x)−g′(x)\)
7. \(\quad \dfrac{d}{dx}\left(\dfrac{f(x)}{g(x)}\right)=\dfrac{g(x)f′(x)−f(x)g′(x)}{(g(x))^2}\)
8. \(\quad \dfrac{d}{dx}\left[f(g(x))\right]=f′(g(x))·g′(x)\)
Trigonometric Functions
9. \(\quad \dfrac{d}{dx}\left(\sin x\right)=\cos x\)
10. \(\quad \dfrac{d}{dx}\left(\tan x\right)=\sec^2x\)
11. \(\quad \dfrac{d}{dx}\left(\sec x\right)=\sec x\tan x\)
12. \(\quad \dfrac{d}{dx}\left(\cos x\right)=−\sin x\)
13. \(\quad \dfrac{d}{dx}\left(\cot x\right)=−\csc^2x\)
14. \(\quad \dfrac{d}{dx}\left(\csc x\right)=−\csc x\cot x\)
Inverse Trigonometric Functions
15. \(\quad \dfrac{d}{dx}\left(\sin^{-1}x\right)=\dfrac{1}{\sqrt{1−x^2}}\)
16. \(\quad \dfrac{d}{dx}\left(\tan^{-1}x\right)=\dfrac{1}{1+x^2}\)
17. \(\quad \dfrac{d}{dx}\left(\sec^{-1}x\right)=\dfrac{1}{|x|\sqrt{x^2−1}}\)
18. \(\quad \dfrac{d}{dx}\left(\cos^{-1}x\right)=\dfrac{-1}{\sqrt{1−x^2}}\)
19. \(\quad \dfrac{d}{dx}\left(\cot^{-1}x\right)=\dfrac{-1}{1+x^2}\)
20. \(\quad \dfrac{d}{dx}\left(\csc^{-1}x\right)=\dfrac{-1}{|x|\sqrt{x^2−1}}\)
Exponential and Logarithmic Functions
21. \(\quad \dfrac{d}{dx}\left(e^x\right)=e^x\)
22. \(\quad \dfrac{d}{dx}\left(\ln|x|\right)=\dfrac{1}{x}\)
23. \(\quad \dfrac{d}{dx}\left(b^x\right)=b^x\ln b\)
24. \(\quad \dfrac{d}{dx}\left(\log_bx\right)=\dfrac{1}{x\ln b}\)
Hyperbolic Functions
25. \(\quad \dfrac{d}{dx}\left(\sinh x\right)=\cosh x\)
26. \(\quad \dfrac{d}{dx}\left(\tanh x\right)=\text{sech}^2 \,x\)
27. \(\quad \dfrac{d}{dx}\left(\text{sech} x\right)=−\text{sech} \,x\tanh x\)
28. \(\quad \dfrac{d}{dx}\left(\cosh x\right)=\sinh x\)
29. \(\quad \dfrac{d}{dx}\left(\coth x\right)=−\text{csch}^2 \,x\)
30. \(\quad \dfrac{d}{dx}\left(\text{csch} \,x\right)=−\text{csch} x\coth x\)
Inverse Hyperbolic Functions
31. \(\quad \dfrac{d}{dx}\left(\sinh^{-1}x\right)=\dfrac{1}{\sqrt{x^2+1}}\)
32. \(\quad \dfrac{d}{dx}\left(\tanh^{-1}x\right)=\dfrac{1}{1-x^2}\quad (|x|<1)\)
33. \(\quad \dfrac{d}{dx}\left(\text{sech}^{-1}\,x\right)=\dfrac{-1}{x\sqrt{1-x^2}}\quad (0<x<1)\)
34. \(\quad \dfrac{d}{dx}\left(\cosh^{-1}x\right)=\dfrac{1}{\sqrt{x^2-1}}\quad (x>1)\)
35. \(\quad \dfrac{d}{dx}\left(\coth^{-1}x\right)=\dfrac{1}{1-x^2}\quad (|x|>1)\)
36. \(\quad \dfrac{d}{dx}\left(\text{csch}^{−1}\,x\right)=\dfrac{-1}{|x|\sqrt{1+x^2}}\quad (x≠0)\)