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Calculus Made Easy

We see that, since

${\frac {dx}{dy}}\times {\frac {dy}{dx}}=1$ and ${\frac {dx}{dy}}={\frac {1}{y}}\times {\frac {1}{\log _{\epsilon }a}},\quad {\frac {1}{y}}\times {\frac {dy}{dx}}=\log _{\epsilon }a$ .

We shall find that whenever we have an expression such as $\log _{\epsilon }y=$ a function of $x$ , we always have ${\dfrac {1}{y}}\,{\dfrac {dy}{dx}}=$ the differential coefficient of the function of $x$ , so that we could have written at once, from $\log _{\epsilon }y=x\log _{\epsilon }a$ ,

${\frac {1}{y}}\,{\frac {dy}{dx}}=\log _{\epsilon }a\quad {\text{and}}\quad {\frac {dy}{dx}}=a^{x}\log _{\epsilon }a.$

Let us now attempt further examples.

Examples.

(1) $y=\epsilon ^{-ax}$ . Let $-ax=z$ ; then $y=\epsilon ^{z}$ .

${\frac {dy}{dz}}=\epsilon ^{z}$ ; ${\frac {dz}{dx}}=-a$ ; hence ${\frac {dy}{dx}}=-a\epsilon ^{-ax}$ .

Or thus:

$\log _{\epsilon }y=-ax;\quad {\frac {1}{y}}\,{\frac {dy}{dx}}=-a;\quad {\frac {dy}{dx}}=-ay=-a\epsilon ^{-ax}$ .

(2) $y=\epsilon ^{\frac {x^{2}}{3}}$ . Let ${\dfrac {x^{2}}{3}}=z$ ; then $y=\epsilon ^{z}$ .

${\frac {dy}{dz}}=\epsilon ^{z};\quad {\frac {dz}{dx}}={\frac {2x}{3}};\quad {\frac {dy}{dx}}={\frac {2x}{3}}\,\epsilon ^{\frac {x^{2}}{3}}$ .

Or thus:

$\log _{\epsilon }y={\frac {x^{2}}{3}};\quad {\frac {1}{y}}\,{\frac {dy}{dx}}={\frac {2x}{3}};\quad {\frac {dy}{dx}}={\frac {2x}{3}}\,\epsilon ^{\frac {x^{2}}{3}}$ .