342
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
the exponent hn tends to 0, so that on replacing the factor Cn by l/hn one
will find a sequence Vn which tends to log2. Now Un = cnhnvn , and since
enhn is a rational fraction in n whose terms of highest degree have ratio 1/3,
one concludes that lim Un = log(2)/3. Confronted by this kind of challenge, a
thoughtless or a somewhat slow young man (there were no young girls) might
be deprived of the ineffable privilege of wearing a cocked hat and a sword On
the Champs Elysees On national holidays9, not to mention the incalculable
consequences to the course of his career.
8 - Derivatives of exponential functions, powers and logarithms
Formula (7.8) also allows us to calculate the derivatives of the other functions
defined above:
Theorem 6. The exponential, logarithmic and power functions are indefinitely differentiable, and:
(8.1)
(ax)' = log(a).a x ,
(aX)" = log(a)2.a x , etc.
(8.2)
log:(x) = l/log(a)x,
log~(x) = -1/log(a)x2, etc.
(8.3)
(X s ), = sx s - 1 ,
(X S )" = s(s _1)x S - 2 , etc.
The exponential functions have been dealt with in the preceding nO. From
this one can settle the case of any logarithmic function g = loga.
First of all, it is easy to guess a priori, up to a few details, the value of
its derivative, assuming we have proved its existence. Let us start from the
relation
g(ax) = g(x) + g(a)
and, for a fixed and x varying, let us calculate the derivatives of the two sides.
That of the left hand side is ag' (ax) and that of the right hand side is g' (x).
So ag'(ax) = g'(x) for all a and x, whence ag'(a) = g'(l); since a is arbitrary
we can write this result in the form
(8.4)
g'(x) = g'(l)jx.
One might also, as in the preceding case, remark that
(8.5)
g(x + h) - g(x) = g(l + h/x) = g'(l)h/x + o(h).
But we still have to justify the existence of the derivative. To do this we use
rule (D 5) of Chap. III, nO 15, concerning the derivatives of inverse functions.
9 Speaking of what he calls his "incorporation" into the Ecole poly technique in
1951, a disabused nucleocrat wrote: "This was the day that I fully understood
with horror that [Poly technique) was a boarding school under military discipline". Yves Girard, Un neutron entre les dents (Paris, Ed. Rive droite, 1997),
p.18.
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
the exponent hn tends to 0, so that on replacing the factor Cn by l/hn one
will find a sequence Vn which tends to log2. Now Un = cnhnvn , and since
enhn is a rational fraction in n whose terms of highest degree have ratio 1/3,
one concludes that lim Un = log(2)/3. Confronted by this kind of challenge, a
thoughtless or a somewhat slow young man (there were no young girls) might
be deprived of the ineffable privilege of wearing a cocked hat and a sword On
the Champs Elysees On national holidays9, not to mention the incalculable
consequences to the course of his career.
8 - Derivatives of exponential functions, powers and logarithms
Formula (7.8) also allows us to calculate the derivatives of the other functions
defined above:
Theorem 6. The exponential, logarithmic and power functions are indefinitely differentiable, and:
(8.1)
(ax)' = log(a).a x ,
(aX)" = log(a)2.a x , etc.
(8.2)
log:(x) = l/log(a)x,
log~(x) = -1/log(a)x2, etc.
(8.3)
(X s ), = sx s - 1 ,
(X S )" = s(s _1)x S - 2 , etc.
The exponential functions have been dealt with in the preceding nO. From
this one can settle the case of any logarithmic function g = loga.
First of all, it is easy to guess a priori, up to a few details, the value of
its derivative, assuming we have proved its existence. Let us start from the
relation
g(ax) = g(x) + g(a)
and, for a fixed and x varying, let us calculate the derivatives of the two sides.
That of the left hand side is ag' (ax) and that of the right hand side is g' (x).
So ag'(ax) = g'(x) for all a and x, whence ag'(a) = g'(l); since a is arbitrary
we can write this result in the form
(8.4)
g'(x) = g'(l)jx.
One might also, as in the preceding case, remark that
(8.5)
g(x + h) - g(x) = g(l + h/x) = g'(l)h/x + o(h).
But we still have to justify the existence of the derivative. To do this we use
rule (D 5) of Chap. III, nO 15, concerning the derivatives of inverse functions.
9 Speaking of what he calls his "incorporation" into the Ecole poly technique in
1951, a disabused nucleocrat wrote: "This was the day that I fully understood
with horror that [Poly technique) was a boarding school under military discipline". Yves Girard, Un neutron entre les dents (Paris, Ed. Rive droite, 1997),
p.18.