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Derivative formulas

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The derivative of any polynomial function of one variable is easily obtained. If c \in \mathbb R (or a constant function) and f,g:D \rightarrow \mathbb R are both differentiable on some set D', then so are cf, f+g, -f, and fg. If, in addition, g is nonzero on D', then \frac 1 g (and also \frac f g) are differentiable on D'. Also, if f is differentiable on g\left(D'\right), then f \circ g is differentiable on D'. For the trivial case of f(x) = a, for some constant a (a degree 0 polynomial):

a'=0 [Proof]

For any real number r:

(x^r)'=rx^{r-1} [Proof]

Which covers any single variable polynomial function. Derivatives of non-polynomial functions require additional rules.

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

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)}

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