The exponential function is a function of the form f(x)=e^x . Euler's number, e , is the base of the natural logarithm.

Sometimes, in general, the term exponential function can refer to functions of the form f(x)=ka^x , where a,k are constants. The variable a here is the base.

Properties of exponential functions

  • a^0=1
  • \lim_{x\to\infty}a^x=0\quad\text{if }a<1
  • \lim_{x\to\infty}a^x=\infty\quad\text{if }a>1
  • \frac{d}{dx}(a^x)=a^x\ln(a)
  • \frac{d}{dx}(e^x)=e^x
  • \int a^xdx=\dfrac{a^x}{\ln(a)}+C
  • \int e^xdx= e^x+C
  • \int\limits_{-\infty}^0e^xdx=1
  • \int\limits_0^1e^xdx=e

See also

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