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For any constant a:

$ a'=0 $

For any real number r:

$ (x^r)'=rx^{r-1} $ (Power rule)

For any real-valued differentiable functions $ f(x) $ and $ g(x) $:

  • $ a\,f'(x)=a\,f'(x) $
  • $ (f(x)+g(x))'=f'(x)+g'(x) $
  • $ \left(cf\right)' = c\left(f'\right) $ (Constant rule)
  • $ \left(-f\right)' = -\left(f'\right) $
  • $ \left(fg\right)' = f'g+fg' $ (Product rule)
  • $ \left( \frac 1 g \right)' = -\frac {g'} {g^2} $
  • $ \left( \frac f g \right)' = \frac {f'g-fg'}{g^2} $ (Quotient rule)
  • $ \left(f\circ g\right)' = \left(f'\circ g\right)g' $ (Chain rule)

Trigonometric functions:

  • $ (\sin(x))' = \cos(x) $
  • $ (\cos(x))' = -\sin(x) $
  • $ (\tan(x))' = \sec^2(x) $
  • $ (\csc(x))' = -\csc(x) \cot(x) $
  • $ (\sec(x))' = \sec(x) \tan(x) $
  • $ (\cot(x))' = -\csc^2(x) $
  • $ (\arcsin(x))' = \frac{1}{\sqrt{1-x^2}} $
  • $ (\arccos(x))' = -\frac{1}{\sqrt{1-x^2}} $
  • $ (\arctan(x))' = \frac{1}{1+x^2} $
  • $ (\arcsec(x))' = \frac{1}{|x|\sqrt{x^2-1}} $
  • $ (\arccsc(x))' = -\frac{1}{|x|\sqrt{x^2-1}} $
  • $ (\arccot(x))' = -\frac{1}{x^2+1} $

Logarithmic and exponential functions:

  • $ (e^x)'=e^x $
  • $ (a^x)'=a^x \ln(a) $
  • $ (\ln x)'=\frac{1}{x} $
  • $ (\log_{a}x)'=\frac{1}{x \ln(a)} $

See alsoEdit