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Physics: Problems and Solutions

Table of derivatives

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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

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