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12.A: Table of Derivatives

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  • Page ID
    157037
    • Gilbert Strang & Edwin “Jed” Herman
    • OpenStax

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    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. \(\quad \dfrac{d}{dx}\left(\text{csch}^{−1}\,x\right)=\dfrac{-1}{|x|\sqrt{1+x^2}}\quad (x≠0)\)

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