Page:Calculus Made Easy.pdf/165

 and therefore

when $$x=x,\quad y=(2.718281 \text{ etc}.)^x$$; that is, $$y=\epsilon^x,$$ thus finally demonstrating that

[Note.–How to read exponentials. For the benefit of those who have no tutor at hand it may be of use to state that $$\epsilon^x$$ is read as “epsilon to the eksth power;” or some people read it “exponential eks.” So $$\epsilon^{pt}$$ is read “epsilon to the pee-teeth-power” or “exponential pee tee.” Take some similar expressions:–Thus, $$\epsilon^{-2}$$ is read “epsilon to the minus two power” or “exponential minus two.” $$\epsilon^{-ax}$$ is read “epsilon to the minus ay-eksth” or “exponential minus ay-eks.”]

Of course it follows that $$\epsilon^y$$ remains unchanged if differentiated with respect to $$y$$. Also $$\epsilon^{ax}$$, which is equal to $$(\epsilon a)^x$$, will, when differentiated with respect to $$x$$, be $$a\epsilon^{ax}$$, because $$a$$ is a constant.

Natural or Naperian Logarithms.

Another reason why $$\epsilon$$ is important is because it was made by Napier, the inventor of logarithms, the basis of his system. If $$y$$ is the value of $$\epsilon^x$$, then $$x$$ is the logarithm, to the base $$\epsilon$$, of $$y$$. Or, if

The two curves plotted in Fig. 38 and 39 represent these equations.