Formulas and Tables

Calculus

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 \(u{\left( x \right)^{v\left( x \right)}}\)
    \({\large\frac{d}{{dx}}\normalsize} \left( {{u^v}} \right) =\) \(v{u^{v – 1}} \cdot {\large\frac{{du}}{{dx}}\normalsize} \,+\) \({u^v}\ln u \cdot {\large\frac{{dv}}{{dx}}\normalsize}\)

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