We see that, since
d
x
d
y
×
d
y
d
x
=
1
{\displaystyle {\frac {dx}{dy}}\times {\frac {dy}{dx}}=1}
and
d
x
d
y
=
1
y
×
1
log
ϵ
a
,
1
y
×
d
y
d
x
=
log
ϵ
a
{\displaystyle {\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
ϵ
y
=
{\displaystyle \log _{\epsilon }y=}
a function of
x
{\displaystyle x}
, we always have
1
y
d
y
d
x
=
{\displaystyle {\dfrac {1}{y}}\,{\dfrac {dy}{dx}}=}
the differential coefficient of the function of
x
{\displaystyle x}
, so that we could have written at once, from
log
ϵ
y
=
x
log
ϵ
a
{\displaystyle \log _{\epsilon }y=x\log _{\epsilon }a}
,
1
y
d
y
d
x
=
log
ϵ
a
and
d
y
d
x
=
a
x
log
ϵ
a
.
{\displaystyle {\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
=
ϵ
−
a
x
{\displaystyle y=\epsilon ^{-ax}}
. Let
−
a
x
=
z
{\displaystyle -ax=z}
; then
y
=
ϵ
z
{\displaystyle y=\epsilon ^{z}}
.
Or thus:
log
ϵ
y
=
−
a
x
;
1
y
d
y
d
x
=
−
a
;
d
y
d
x
=
−
a
y
=
−
a
ϵ
−
a
x
{\displaystyle \log _{\epsilon }y=-ax;\quad {\frac {1}{y}}\,{\frac {dy}{dx}}=-a;\quad {\frac {dy}{dx}}=-ay=-a\epsilon ^{-ax}}
.
(2)
y
=
ϵ
x
2
3
{\displaystyle y=\epsilon ^{\frac {x^{2}}{3}}}
. Let
x
2
3
=
z
{\displaystyle {\dfrac {x^{2}}{3}}=z}
; then
y
=
ϵ
z
{\displaystyle y=\epsilon ^{z}}
.
d
y
d
z
=
ϵ
z
;
d
z
d
x
=
2
x
3
;
d
y
d
x
=
2
x
3
ϵ
x
2
3
{\displaystyle {\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
ϵ
y
=
x
2
3
;
1
y
d
y
d
x
=
2
x
3
;
d
y
d
x
=
2
x
3
ϵ
x
2
3
{\displaystyle \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}}}
.