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Calculus

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Table of First Order Derivatives

  • Functions: \(f\), \(y\), \(u\), \(v\)
    Argument (independent variable): \(x\)
    Natural number: \(n\)
    Real numbers: \(C\), \(a\), \(b\), \(c\)
    1. Derivative of a constant
      \({\large\frac{d}{{dx}}\normalsize}\left( C \right) = 0\)
    2. Derivative of the function \(y = x\)
      \({\large\frac{d}{{dx}}\normalsize}\left( x \right) = 1\)
    3. Derivative of a linear function
      \({\large\frac{d}{{dx}}\normalsize}\left({ax + b}\right) = a\)
    4. Derivative of a quadratic function
      \({\large\frac{d}{{dx}}\normalsize}\left({ax^2 + bx + c}\right) =\) \( {2ax + b}\)
    5. Derivative of the power function
      \({\large\frac{d}{{dx}}\normalsize}\left({x^n}\right) =\) \( {nx^{n – 1}}\)
    6. Derivative of the power function with a negative exponent
      \({\large\frac{d}{{dx}}\normalsize}\left({x^{-n}}\right) = – {\large\frac{n}{{{x^{n + 1}}}}\normalsize}\)
    7. Derivative of the reciprocal function
      \({\large\frac{d}{{dx}}\normalsize}\left( {{\large\frac{1}{x}}\normalsize} \right) = – {\large\frac{1}{{{x^2}}}\normalsize}\)
    8. Derivative of the square root function
      \({\large\frac{d}{{dx}}\normalsize} \left( {\sqrt x } \right) = {\large\frac{1}{{2\sqrt x }}\normalsize}\)
    9. Derivative of a root
      \({\large\frac{d}{{dx}}\normalsize}\left( {\sqrt[n]{x}} \right) = {\large\frac{1}{{n\sqrt[n]{{{x^{n – 1}}}}}}\normalsize}\)
    10. Derivative of the logarithmic function
      \({\large\frac{d}{{dx}}\normalsize}\left( {{{\log }_a}x} \right) =\) \( {\large\frac{1}{{x\ln x}}\normalsize},\) \(a \gt 0,\) \(a \ne 1.\)
    11. Derivative of the natural logarithm
      \({\large\frac{d}{{dx}}\normalsize} \left( {\ln x} \right) = {\large\frac{1}{x}\normalsize}\)
    12. Derivative of the exponential function with base a
      \({\large\frac{d}{{dx}}\normalsize} \left( {{a^x}} \right) = {a^x}\ln a,\) \(a \gt 0,\) \(a \ne 1.\)
    13. Derivative of the exponential function with base e
      \({\large\frac{d}{{dx}}\normalsize} \left( {{e^x}} \right) = {e^x}\)
    14. Derivative of the sine function
      \({\large\frac{d}{{dx}}\normalsize} \left( {{\sin x}} \right) = {\cos x}\)
    15. Derivative of the cosine function
      \({\large\frac{d}{{dx}}\normalsize} \left( {{\cos x}} \right) = {-\sin x}\)
    16. Derivative of the tangent function
      \({\large\frac{d}{{dx}}\normalsize} \left( {\tan x} \right) =\) \({\large\frac{1}{{{{\cos }^2}x}}\normalsize} =\) \( {\sec ^2}x\)
    17. Derivative of the cotangent function
      \({\large\frac{d}{{dx}}\normalsize} \left( {\cot x} \right) =\) \(-{\large\frac{1}{{{{\sin }^2}x}}\normalsize} =\) \({-\csc ^2}x\)
    18. Derivative of the secant function
      \({\large\frac{d}{{dx}}\normalsize} \left( {\sec x} \right) =\) \( \tan x \cdot \sec x\)
    19. Derivative of the cosecant function
      \({\large\frac{d}{{dx}}\normalsize} \left( {\csc x} \right) =\) \( {-\cot x} \cdot \csc x\)
    20. Derivative of the inverse sine function
      \({\large\frac{d}{{dx}}\normalsize} \left( {\arcsin x} \right) =\) \({\large\frac{1}{{\sqrt {1 – {x^2}} }}\normalsize}\)
    21. Derivative of the inverse cosine function
      \({\large\frac{d}{{dx}}\normalsize} \left( {\arccos x} \right) =\) \( -{\large\frac{1}{{\sqrt {1 – {x^2}} }}\normalsize}\)
    22. Derivative of the inverse tangent function
      \({\large\frac{d}{{dx}}\normalsize} \left( {\arctan x} \right) =\) \({\large\frac{1}{{1 + {x^2}}}\normalsize}\)
    23. Derivative of the inverse cotangent function
      \({\large\frac{d}{{dx}}\normalsize} \left( {\text {arccot }x} \right) =\) \(-{\large\frac{1}{{1 + {x^2}}}\normalsize}\)
    24. Derivative of the inverse secant function
      \({\large\frac{d}{{dx}}\normalsize}\left( {\text {arcsec }x} \right) =\) \({\large\frac{1}{{\left| x \right|\sqrt {{x^2} – 1} }}\normalsize}\)
    25. Derivative of the inverse cosecant function
      \({\large\frac{d}{{dx}}\normalsize}\left( {\text {arccsc }x} \right) =\) \( -{\large\frac{1}{{\left| x \right|\sqrt {{x^2} – 1} }}\normalsize}\)
    26. Derivative of the hyperbolic sine function
      \({\large\frac{d}{{dx}}\normalsize}\left( {\sinh x} \right) = \cosh x\)
    27. Derivative of the hyperbolic cosine function
      \({\large\frac{d}{{dx}}\normalsize}\left( {\cosh x} \right) = \sinh x\)
    28. Derivative of the hyperbolic tangent function
      \({\large\frac{d}{{dx}}\normalsize} \left( {\tanh x} \right) =\) \({\large\frac{1}{{{{\cosh }^2}x}}\normalsize} =\) \({{\text {sech}}^2}x\)
    29. Derivative of the hyperbolic cotangent function
      \({\large\frac{d}{{dx}}\normalsize} \left( {\coth x} \right) =\) \(-{\large\frac{1}{{{{\sinh }^2}x}}\normalsize} =\) \(-{{\text {csch}}^2}x,\) \(x \ne 0.\)
    30. Derivative of the hyperbolic secant function
      \({\large\frac{d}{{dx}}\normalsize} \left( {\text {sech }x} \right) =\) \( – {\text {sech }x} \cdot \tanh x\)
    31. Derivative of the hyperbolic cosecant function
      \({\large\frac{d}{{dx}}\normalsize}\left( {\text {csch }x} \right) =\) \( – {\text {csch }x} \cdot \coth x,\) \(x \ne 0.\)
    32. Derivative of the inverse hyperbolic sine function
      \({\large\frac{d}{{dx}}\normalsize}\left( {\text {arcsinh }x} \right) =\) \({\large\frac{1}{{\sqrt {{x^2} + 1} }}\normalsize}\)
    33. Derivative of the inverse hyperbolic cosine function
      \({\large\frac{d}{{dx}}\normalsize}\left( {\text {arccosh }x} \right) =\) \({\large\frac{1}{{\sqrt {{x^2} – 1} }}\normalsize},\) \(x \gt 1.\)
    34. Derivative of the inverse hyperbolic tangent function
      \({\large\frac{d}{{dx}}\normalsize}\left( {\text {arctanh }x} \right) =\) \({\large\frac{1}{{1 – {x^2}}}\normalsize},\) \(\left| x \right| \lt 1.\)
    35. Derivative of the inverse hyperbolic cotangent function
      \({\large\frac{d}{{dx}}\normalsize}\left( {\text {arccoth }x} \right) =\) \(-{\large\frac{1}{{{x^2} – 1}}\normalsize},\) \(\left| x \right| \gt 1.\)
    36. Derivative of the function \(u{\left( x \right)^{v\left( x \right)}}\)
      \({\large\frac{d}{{dx}}\normalsize} \left( {{u^v}} \right) =\) \(v{u^{v – 1}} \cdot {\large\frac{{du}}{{dx}}\normalsize} \,+\) \({u^v}\ln u \cdot {\large\frac{{dv}}{{dx}}\normalsize}\)