0:00:00.367,0:00:08.932 Euler's number and the exponential function based on that number are of vital importance in the theory of differential equations. 0:00:08.933,0:00:16.065 Whenever we are dealing with exponential growth or exponential decay, and whenever we deal with oscillations, 0:00:16.067,0:00:22.099 Euler's number plays a big role. This is a brief and informed introduction to that topic, for which we need a minimum 0:00:22.100,0:00:29.166 amount of prerequisites. In particular, you should know that five to the first power equals five. 0:00:29.167,0:00:38.632 Five to the zeroth power equals one. Five to the minus second power equals one over 25. 0:00:38.633,0:00:43.599 Five to the power of one half equals the square root of five. 0:00:43.600,0:00:53.566 Five to the third power times five to the fourth power equals five to the power of three plus four--seven. 0:00:53.567,0:01:01.566 And five to the third power to the fourth power equals five to the power of 12--three times four. 0:01:01.567,0:01:04.932 This is what you need to know about powers of numbers. 0:01:04.933,0:01:13.032 In addition, you should have an idea about derivatives in this D, DX notation--for instance, the derivate of X squared 0:01:13.033,0:01:20.333 with respect to X equals 2X. And you should have an idea of how this is connected with the slope of the tangent line.