Page:Calculus Made Easy.pdf/178

 or $$\log_\epsilon y = \log_\epsilon b - ax \log_\epsilon \epsilon = \log_\epsilon b - ax$$,

and, differentiating, $$\frac{1}{y}\, \frac{dy}{dx} = -a$$;

hence $$\frac{dy}{dx} = b\epsilon^{-ax} \times (-a) = -ay$$; or, in words, the slope of the curve is downward, and proportional to $$y$$ and to the constant $$a$$.

We should have got the same result if we had taken the equation in the form

The Time-constant. In the expression for the “die-away factor” $$\epsilon^{-at}$$, the quantity $$a$$ is the reciprocal of another quantity known as “the time-constant,” which we may denote by the symbol $$T$$. Then the die-away factor will be written $$\epsilon^{-\frac{t}{T}}$$; and it will be seen, by making $$t=T$$ that the meaning of $$T$$ (or of $$\dfrac{1}{a}$$) is that this is the length of time which it takes for the original quantity (called $$\theta_0$$ or $$Q_0$$ in the preceding instances) to die away $$\dfrac{1}{\epsilon}$$th part—that is to $$0.3678$$—of its original value.