FANDOM


The derivative of any polynomial function of one variable is easily obtained. If $ c\in\R $ (or a constant function) and $ f,g:D\to\R $ are both differentiable on some set $ D' $, then so are $ c\cdot f $, $ f+g $, and $ f\cdot g $. 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$ r\in\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),g(x) $:

  • The definition of the derivative:
$ f'(x) = \lim_{h \rightarrow 0} \dfrac{f(x+h) -f(x)}{h} $
  • $ (a\cdot f(x))'=a\cdot f'(x) $[Proof]
  • $ (f(x)\pm g(x))'=f'(x)\pm g'(x) $[Proof]
  • $ (-f(x))'=-f'(x) $
  • $ \left(\frac{f(x)}{g(x)}\right)'=\frac {f'(x)g(x)-f(x)g'(x)}{g(x)^2} $(Quotient rule)
  • $ \left(\frac{1}{g(x)}\right)'=-\frac{g'(x)}{g(x)^2} $
  • $ (f\circ g)'=\bigl(f(g(x))\bigr)'=(f'(g))\cdot g'(x) $(Chain rule)

Trigonometric functions:

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

Logarithmic and exponential functions:

  • $ \frac{d}{dx}(e^x)=e^x $
  • $ \frac{d}{dx}(a^x)=a^x\ln(a) $
  • $ \frac{d}{dx}(\ln(x))=\frac{1}{x} $[Proof]
  • $ \frac{d}{dx}(\log_a(x))=\frac{1}{\ln(a)x} $
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