# Electronic library

I. F. Quinby Horatio Nelson Robinson.

# A new treatise on the elements of the differential and integral calculus

. (page 3 of 22)
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is equal to the sine of the arc.

Let y = vers, a; := 1 — cos, x, then we have

dy d. vers, x d. (I — cos. x) d. cos. x

~f = J = — ^^ -J ' 1= ;, — = sin. x.

ax ' dx ax ax

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40 DIFFERENTIAL CALCULUS.

39* The circular functions whose differential co-eflScients

have been thus far found are called direct circular functions.

Since the tangent, cotangent, secant, and cosecant may all

be expressed under a fractional form in terms of the sine and

cosine, their differential co-efficients could have been found

by the rule of Art. 22. Thus : —

sin. a?

1st, y =Li7xx\.x=.

^ cos. a;

dy __ co s.'^ X + sin.^ x __ 1
dx COS.- X cos.'^ X

COS. X

2d, yz=cot. 0;=-^

' ^ Sin. X

dy _ sin.^ a; -|- cos.^ a? _ 1

dx sin.^ X sin.'* x

3d,

COS. X

dy sin. x ,

-/ = = tan. X sec, x.

dx cos.^ X

4th, y = cosec. x =

sin. X

dy COS. X ,

y- = ^.— o— = cot. X cosec. X.

dx sin.2 X

The other forms frequently given to the differential co-effi-
cients of the direct circular functions will be readily recognized
by the student familiar with the elementary principles of trig-
onometry.

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SECTION III.

DIFFERENTIAL CO-EFFICIENTS OF INVERSE FUNCTIONS, FUNCTIONS
OF FUNCTIONS, AND COMPLEX FUNCTIONS OF A SINGLE VARI-
ABLE.

40- The inverse circular functions are those in which the
sine, cosine, tangent, <tc,, are taken as the independent varia-
bles, the arcs being the functions. They are written y =
sin*"^ Xjy=- cos.~^ ^> ^ == tan.~^ x, &c.; and are read y equal to
the arc of which x is the sine, cosine, tangent, <fec. These
functions are sometimes written y = arc sin. x^y =. arc tan. cc,
&c., and also y = arc (sin. z=ix),y = arc (tan. = x), &c. ; but
the first notation, being the shortest, and that generally adopted,
will be uniformly used in what follows.

41. If we have y = g)(a;), then the differential co-eflScient
of a;, regarded as a function of y, is the reciprocal of the differ-
ential coefficient of y regarded as a function of cc.

That is, if y=iq}{x), then x must be some function of y,

such as a; = \f;{y) ; whence ^- = cp^ (x), =z\p^ (y) : and, ac-
cording to the principle enunciated, we should have
dx .. . 1

As an example, take the equation

y = g) (a:) = x^ + 2x — 3 ;
from which we get

g=.(«+i,

41

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42 DIFFERENTIAL CALCULUS.

Solving the equation with respect to x, we have

whence -,- =: ± - — — ; but zb x/^JTl — t- -l 1 •

therefore ^^ = — — —— ,

dy 2(x+l)'

which accords with the theorem ; and we will now prove that
what holds in this particular case is true for all cases.

Let y=zcp(x) ... (1) be the given function; and since,
from tlie nature of equations, x must also be some function of
yj suppose x=inf(y) . . . (2) to be that function.

If, in Eq. 1, x receives the increment aXj y will receivo a
corresponding increment Ay : therefore

y + Ay = (r{^ + ^oc) . . . (3).

Now, Eqs. 1 and 2 are but different forms of the expression
of a certain relation between the variables x and y ; and what-
ever values of y in Eq. 1 result from an assigned value to
X, if one of these values of y be assigned to y in Eq. 2, then,
among the different resulting values of a;, one at least must be
the value assigned to x in Eq. 1.

It is therefore •proper to assume that x and y have tho
same values in Eq. 2 that they have respectively in Eq. 1.
Change, then, in Eq. 2, y into y + Ay, and x into a;-|-Air,
these symbols having the same values that they have in Eq.
3: hence

x + Ax = xp{y + Ay) . . . (4).

From Eqs. 1 and 3, we have

A y __ (p{x-\- ax') — q}(x)

AX AX

(5);

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DIFFERENTIAL COEFFICIENTS. 43

from Eqs. 2 and 4,

Ay Ay

multiplying Eqs. 5 and 6, member by member^ then

AX Ay AX Ay

By the preceding remarks, ^ x — = 1^ J ^"^> i'^ tl^®

A a; Ay

second member of Eq. 7, the first factor at the limit becomes
_- = g;^(a;); and the second factor, =rt/''(y): hence

4j?. If we have

y = y^{z) ... (1)

z^^ix) ... (2):
then y is a function of x ; for, by substituting in the first of
these equations the expression for z in tho second, y becomes
an explicit function of x. Suppose this to be denoted by

y=f(x) ... (3):

Now, if a;, in Eq. 2, receive the increment A a;, z will take
the increment A z ; and, in consequence of this increment of z,
y in Eq. 1 will become y -\- Ay : hence wo should have, from
Eqs. 1 and 2,

y + Ay = v;(25 + A2) ... (4)
2 -f A« = g'(JJ + Aa;) . . . (5);
also, from Eq. 3,

y + Ay=/(a; + Ax) . . . (6).

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44 DIFFERENTIAL CALCULUS,

From the mode of dependence of the variables, we may
assume that the, symbols x, y, z, ^ Xj a y, A Zj have respec-
tively the same values in all of the preceding equations in
which they occur. Subtracting Eq. 3 from Eq. 6, member
from member, and dividing both members of the result by A x,
we have

AX AX ' ' ' \ )9

similarly, from Eqs. 1 and 4,

Ay _ W{ Z + A Z)-XV {Z) ,

AZ AZ ' * ' \ Jf

and, from Eqs. 2 and 5,

AZ_ (f{x + Ax)-(f{x ) ^

AX AX • • • I /•

Multiplying Eqs. 8 and 9, member by member, we have,
because the symbols are supposed to have the same values
throughout.

Ay A_z _Ay _ u^{z + Az) — w(z) (p{x + Ax) — q{x ) ^ ^ .^^..

A Z AX AX AZ AX

equating the second members of Eqs. 7 and 10,

f{x + Ax )—f (x) __ XP{Z + AZ) — X V{Z) ^ cp(x + Ax) - (p(x\
AX AZ AX

Whence, passing to the limit,

/'{x) = xp'{z)<r'{x),

dy dij dz

'''^ dx dz dx

Ex. 1. . y = z5 + 32 — 5 . . . (1), 2 = 2a; -3 .-. . (2)

^^-=2. + 3, J- = 2, ^^ = ^^^ = 4. +6
dz ax ax dz dx

= 8ic — 6.

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DIFFERENTIAL CO-EFFICIENTS. 45

By placing for 2, in Eq. 1, its value from Eq. 2, we find

dy
y =z ^x^ — 6x — 5, whence ^- =80; — 6 ;

the same result as was found by the first process.

4:3* DiflFerential co-efficients of the inverse circular func-
tions.

1st, Differential co-efficient of y := sin.~^a;.
Since y = sin,~^ic, x = sin,y; and therefore, by Art. 32,
dx
dy'
and therefore, by Art. 41,

dx COS. y \/l — x^

2d, Differential co-efficient of y = cos.~^ x.

Here y = co3.~^a? gives x = cos. y ; therefore. Art. 33,

dx . . ^

- z= — sin. y = =F Vl — a;2;

and therefore, by Art. 41,

'z=cos.y=± Vl— irS'

cLc sin. y y'l x'^

It would be superfluoua to point out tlie necessity for the
signs i> =p, before the differential co-efficients in this and
the preceding case.
3d, Differential co-efficient o{ y = tan.~^ x.
From y = tan.~^ x, we have x = tan. y : therefore
dx 1

^ = ^s7^ = '""-'^=^ + *"^-'^ (Art. 34);

dy 1

and -y- = cos.^y =: 1—7—7 — :^ — (Art. 41).

dx ^ 1+taiL^y

Whence

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46 DIFFERENTI.!!. CALCULUS.

ithj DifTerential co-eflScient o{ y = cot.~^ x.
From y = cot.~^ x, we have x = cot. y : therefore
dx 1

dy sin.2 y

— cosec.^y = _ (1 _|_ cot.^y) (Art. 35).

5th, Differential co-efficient o{ y = sec.~^ x.
Prom y = sec.~^ x, we have x = sec. y : therefore

and ^y ^ c_os^ ^ / — (Art. 41).

ax Bin. y sec- y sin. t^

But sec. y = , hence cos. y = = - ; and

COS. y secy x

1 Va:^ — 1
1 — sin.^ y z=z—s^ sin. y = ± • therefore

dx X \/x'- — i

6th, Differential co-efficient of y = cosec.~^a:.

We shall merely indicate the steps.

dx COS. y . . , . ^ o^x

X = cosec y, -^^ = - ^-^ = - cosec y cot. y (Art. 37):

dy 1.11

-/ = — , sm. y = —=-;

ax cosec. y cot. y cosec y x

7th, Differential co-efficient of y = vers.-^ a;.
Taking a; for the function, we have

X = vers, y = 1 — cos. y ;

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DIFFERENTIAL CO-EFFICIENTS. 47

therefore ^ = sin. y (Art. 38), and f ?^ = -J— (Art. 41):

dy . ^ ^ ' ax sm. y ^ ^

but -^=^ — ^ =ik

sin. y V 1 — C0S.2 y Vl — (1 — vers, y)^

Vl — (1 — ir)2 V2a; — a;2

44. The principle demonstrated in Art. 41 has greatly
simplified the investigation of the formulas expressing the
differential co-efficients of the inverse trigonometrical func-
tions. They may, however, be determined directly, without
the aid of this principle.

We will illustrate the manner in which this may be done
by a single example : —

Let y = sin.~^ cc, then

y + Ay = 8in-i(a; + A);
and Ay — sin."^ {x + li) — sin.~^ x.

The second member of this last equation is the difference
of two arcs whose respective sines are x -\-h and x] and this
difference is, by trigonometry (Plane Trig., Eq. 8), equal to
an arc having for its sine the sine of the first arc multiplied
by the cosine of tlie second, minus the cosine of the first
multiplied by the sine of the second. Expressing the cosines
of these arcs in terms of their sines, we have
Ay = sin.~' {x -f- li) — sin.""^ x

= sin.-i ((x + A) V"!^^ - ^ V [1 - (^ + '0']) :

^y _^Binr' ({x+h)^'r^^^^x^ [1 - (x + hy])

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(

48 DIFFERENTIAL CALCULUS.

Make z = {x + h) ^T^"^- — x^[\-{x + A)^] :

therefore >^ = — . — = _- .

aOc a z h

Now z and h diminish together, and become zero simultane-

sm — ^ z a

oush\ At the limit, — - — = 1, To find what 7 becomes at

the same time, multiply and divide the expression for z by
(x + h)^l — x'' +x^(l — {x + h)A ; then

^ {x + hy{i-x')-x'(i-^{x + jiy)
* h ((^x +h)V'i-~^' + X v[i - (^ + hy]\

__ 2x-^h

^ (x +h)^TZ.~x' + X V(l - (a: + A)A '

Pass to the limit* by making h =1 0, and we have

,, z X 1

lim. ^ =

45. Differential co-efficients of functions of the form y = i*
in which t and 5 are both functions of the same variable x.

Taking the Napierian logarithms of both members of the
equation y = t*j we have ly = sIL By Art. 42, the differ-
ential co-efficient with respect to x of ly is

dJy^l dy^ ^^^^ ^^ ^^^ 20, 42, that of sit is

^ c7.5 c/a- rf.tt dt __j ds 1 dt
ds dx dt dx dx t dx

Now, since the equation ly = sit is true for all values of cc,

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DIFFERENTIAL CO-EFFICIENTS, 49

the diflFerential co-efficients of its two members most also bo
equal: therefore

1 rfy _ , cfo « rf^

"— |=.(4+i|)

='-«l+ - l='-(4+fl)-

4:6. Prom an examination of the particular cases treated in
Arts. 19, 20, 45, we deduce this general rule for finding the
differential co-efficient of any compound function: Differen-
tiate each component function in succession, treating the others
as constant, and take the algebraic sum of the results.

Rules have now been given for the differentiation of all
known forms of algebraic, logarithmic, exponential, and circular
functions of a single variable ; and we have seen, that, in gener-
al, the differential co-efficients of these functions are themselves
functions of the same variable.

4:7' The following exercises are given that the preceding
rules may be impressed, and that the students may become
expert in their application, and familiar with the forms of the
differential co-efficients of simple and complex functions : —

1. y = ax\^^^=^^ax^ (Art. 28).

2. y = abx^ — cx^

^l = Sabx^ — 2cx (Arts. 19, 28).

_ 2ax(b + 2x^)
T ~ (b-x^)-'

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But

and

50 DIFFERENTIAL CALCULUS.

^' y = \^^^+Wx + c= {x + {x + off.

Put z = x-{-{x + c)l'y t]ie:i y ==: zi,

therefore f^- ^ ^3(ar + r)l + l

2(x + (a: + c/^ ) 3(x + c)S

3 V(^+^c)M- 1

~ 6 V^+V^c X V(a: "+ cf

5. y=:/(a; + \/l+.f^). Make2 = x-4-Vl+a?2;

then y = Iz, and ,^ = -j/- -/- .

*^ ' ax dz ax

^z/_^i^ L

'2 a:+Vl + a;^

But 7=zi:zz. i-_=:. (Art. 31).

:md ^=1 + - :^„_^ ^ + Vl + ^- \

^*^ Vl + a:'^ Vl+a;^

therefore ^^ = ^^^ l- _ ^^^^^1+^^ L_.

The utihty of substituting a single symbol to represent a
compHcated expression before differentiation is exemphfied in
this and tlie preceding examples. Oftentimes the labor, both
mental and mechanical, of the mathematician is greatly abridged
by the adoption of suitable artifices.

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DIFFERENTIAL CO-EFFICIENTS, . 51

6. V =^l -—^^^^=^- Multiply both numerator and de-

^ A^l^x' + x

nominator by the numerator, then

Put z = l +2x^^2xV l+~^\ AVhence y = Izj

- dif dy dz ,

dx dz dx

but
and

1

dz 2 i_|_2x-^— 2xVl+cc'^'
dz , - - 2x2

2(l + 2a;- — 2x^1+3?-) . ^.^

di_ 1 2 (l + 23;^-2xVl+^')

dx - 1 + 2x^-2x^1^^^ ^ V'r+^"

7. y = tan.-'

Vl+x'^
3a -X — x'

a(a'^ — 3x'^)

, cZ^ dy dz dy

dx dz dx dz 1+2 "^

1 d\a'-^xy

l-\-z' /?ya'x — x^\ "" {a"- + x-)=^

^2 __ 3 (a^ - X-) (a2 — 3x-) + 6x (Sxa^ — x^)

dx a(a'^ — Sx'^y^

__ 3 (a^ + 2a''x^ + x^) __ 3 {a'^ + x'f ^
~~ a (a' — 3x-)- a(a-^— 3x''^)'^'

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52 DIFFERENTIAL CALCULUS,

therefore ^^.=^1.^= ^lS^H^'I ^ -i(«!±^\

dx dz dx {a^ + xy a(a^ — 3a;2)2»

dy __ 3a
dx^a' + x''
This is as it should be ; for

, Sa^x— x^ ^^ . X*

tan -^ — 7-» ij—><- — Stan -^ - :

a{a^ — 6x-) a

therefore y =:S tan.~"^ -• Make - =^z, then

1 . dy dy dz ,

y = S tan. 2, and j~ ^^ 3" j^— »

CX^ Ct^S CIX

dy i 3 3a« ,,

and

cZz 1 . dy _ 3 a

rfo? a ' ' dx a^-^x'^'

* To prove this, take Eqs. 28 and 33 Plane Trig., and in 28 make 6 = 2a;

then

tan. a -\- tan. 2a

tan. 3a = ; •

1 — tan. a tan.2a

Substituting in the second member of this the value of tan. 2a taken from Eq. 33,

and reducing, we find

tan. 3a = 3 tan, a- tan.Sa
1 — 3tan.*^a

Dividing both numerator and denominator of the fraction

3^-f!

by a^, it becomes — ^—j ; and, comparing this with the formula for the

1—3-^

tangent of 3a, we conclude that it is the expression for the tangent of three timet

the arc of which — is the tangent : therefore

3-—^ 3a^x-x^ ,

3tan.-i- = tan.-i— ^^ 'i_=tan.-l ^^^^^^2)

as was assumed.

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DIFFERENTIAL CO-EFFICIENTS. 53

Q e^(a sin. x — cos. x)

% 1 rf «. / •

~ = o . V "7~ ^ {^ Sin. a; — cos. x)

ax a^ -{-1 ax '

-J- e^{a Bin. x — cos. x) = ae^(a sin. x — cos. x)

+ e"*(a cos. X + sin. a?)
=:(a2 + l)e«'8in.aj;

• •

-/- = e"^ sin. X.
dx

9.

1 . ,6 + asin.a;

y = sin.-^ , . .

Va^— ^2 a + osm.o;

Put

ft + asin.a; ^, 1 .• ,

a + 6sin.a. ;^; theny ^^^^^^ Bin. i«,

nn/l

rfy rfy c?2j

and

rfa? "" dz dx

But

dy

11 11

dz"

>l Va + ftsin.a;/

a -|- 6 sin. a:

Va^ — 6*cos.a;

, rf« _(a_+i^n. a;)acos. ic — (6 + a8in.a;)6cos.aj

dx (« + 6 sin. aj)^

_ (a* — 6^) COS. a; ^
~" (a + 6 sin. aj)'-*

therefore ^ = L_ ^- +^jL"'^ .(^l" ^') «^s- ^

rfa; Va2 — 6^ Va-— ft^cos.a? (a + &sin.a;)2

_ 1

a + 6 sin. a; *

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54 DIFFERENTIAL CALCULUS,

.^^n 1 _, 6 + acos. a; , u ^ j -^

10. If?/=: ^ - r^cos. ^ — '-, J we should find, m

like manner^

dx a + 6 COS. iT '
11. 2, = tan-'(^__^,-.^j.

Put 2 = . ; then y =^ tan. ^ «,

df ?/ _ (Jy <^^- __ 1 ^
^^^ c/^^c/2 c?a;^l + 2^dx

U.i

f^^-^cos.x — g-^sin.a;)(l +e^sin.a;) — g^cos.x(e^cos.a;-f-e^8in.a?)

{\ -\- e^ ^\i\, xf

_ e^ (cos. X — sin. a; — e^)
"" (1 -|- e-^'sin. x)- '

1 1 _ (1 + g^si n. a;f

and

l"+^2— / e-^cos.a; V""(l + e^sin.a;)- + (e^cos.a;)

\ 1 + e^ sin. x)

__ (1 + e-^sin.a^y^
"" l + 2e-^sin.a;+"^*

dy e-^{cos.x — sin. a:— e^)
dx 1 + 26"^ sin. x + e'^

therefore :j^ — i _l Oz,:^ o^r. ^ _l ^2x

12. 2/-^-^,, c?x~'"(r+^^)^ '

13. y = xlx, y^^ = l+ Ix.

dy 2

14. V = ^ cot. X, -,- = . — jr-.

^ ^ dx sin. 2a;

X dy o}

15. y— ^ **

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16.

DIFFERENTIAL CO-EFFICIENTS. 65

"• ^=0" ^=«(-)"(' + ^

^ ~" {l+xf dx ~ (l+x)»-*-i'

19 ^^ ^"-^"" ^I^ __£__.

•or

20. y = Z (6- + e-% ^y = ^^~^ - .

21. ' y=r(a + ir)'^(& + :c)",

^^ = (a + :c)'"-i (h + n;)«-i I m (6 + a;) + n (a + ic) I -

22. y = — 3 tau. a; + a:, /- z= tan.* cc.

o ax

23. y^. ^ ,^ = __ , o'-Vf^:^

x + Vl — x-^ dx Vl—a;^ (1 +2x^/1— icO

24. y=(a* + a!«)tan.-i-, ^^= 2xtan.->- + a.

^ ( ^ ^ M rfx (a + 6x")Z(a + 6x»)

26. y = ltB.n.(''+% \$^ = _J_. •

\4 2/ dx cos.x

o- va + ^ di/ A^ax — a

^~Va + V^' di~2 Vax + x- ( v^« + V^Y

28. y= liS ?y = ^

^ Nl— x' rfx (l_a;)Vi-a;2

90 « - '^ ^y - e "(^-a;)-l-

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56 DIFFERENTIAL CALCULUS.

^Vl+a^ + Vl—o; ^_. 1

^ dx xV

dy _ ny

^ Vnpi— Vl — x^^^-^ xVl— a^*

31. y = ( . f__-Y

dx ajVl— x*

-fa.

32. y = a^-'— , rfx-(y_a;2)i

33. y^e'/'^+lY, ^ = .'__^llll_

\^-l/ rfx (x+l)»(a;-l)*

34. y^ v^r+^ + Vj^^ " rfy^ 2/ 1 \

V 1 + x^ - V 1 — x^ ' dx x\ Vl - xV '

J. y = 8in.~' mx,

^^ (l-m^x*)*'

36. y=:x8in.-'x, - - = sin.~'x4- ^

37. v=:e""~\ ^ _„/«.." X

rfx 1 + X«'

38. y = x""~'- ^y - ^^-K M^ + (1 - ^') ^ sip.-' x \
' ^^ V x(l-x2j* /

39 V _ =» - 8'"-'

x

~ 7

sin.ajf 1 —

Vi

1 — S (x — sin,~^ x) cos X

dy \

. -W - / - -

dx

sin.* a;

a + 6tan
40. y=:l

a — 6 tan

X

"2

— )

X

'2

dy oh

dx'" ^ ^X 7 2 • 2 ^

a- COS.- TT — 0^ sm.^ ^.

2 2

1
41. y=zx'f

1
d^ = xx(\^lx)

dx a?2

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DIFFERENTIAL CO-EFFICIENTS. 57

42. y = e', ^ = e'e*.

43. y = x'', ^ = x''x'(^ + lx + ilxyy

44. y = x', fr^^V^ + ^^ '

dx X

45. y = sin.

„• —1

a? 4" 1 ^y ___

V^ ' rfo; Vl — 2a; — o;*^*

46. y = i^n.-'~^=:,^ = —A=.

Vl-x^ dx Vl — x^

47. y = tan. vT=^

^y _. 1 1 _ __ (sec. \/TZr^y

- dx (cos. s/lZr^f 2 Vni^ "" 2VT^' ■

48. y = tan.-i (wtan. a;), f^ = !?_.^ .

■ X

49. v =
50

y = tan. - ^ ^- ^a^^^)

51. y = tan.-^-^, f = ^.
^ l - x^' dx 1 + x'

52. y = 8in.-VSS:^ 5! = !^^ + cosec.a?^

53. y = sin.-

oo;

rfy g (6 — ca ;^)

cfx "~ (b + ca;2) V6H^"26y=: a^) a;2 + c^o;*'

<^y a?8in.""^a;

54, y = Vl —

x^ sin.""^ a? — a;, -r- =

'.' rfa; \/l —

a;^

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58

DIFFERENTIAL CALCULUS.

rr a; sin. ^x d?/ sin.^^a;

Vl — x'' ' dx {l-x'-)i

, xian.b dy a Man. 6 1

56. y = Sin.~^— - ^rr=-^ - ^=

V a"-— x*^ dx a^ — x'^ {d' — x'^si

57. y = sin.^^

a^ — x' dij _
\b'^ — X'' dx^

b)i

■x^' dx {b'^x'^)Vd^^x

58. y = tau,-'^-\

, . _ ■ COS. X

vr+"cos7

dy^l

X ^^ ^*

59. y = f.n -1 ({<^'^''^ sin^Y f^ ., ja' - h^)^

\ b + acos.x J dx a + 6 cos. a;

60. t/ = sec.-l-— L_^, ''^zzr ^- ,

2^ = tan.-^ 1- , dx-^^

61.

2. y liz: Z ,^' ^ — ' + 2 tan. * ~-^ — -, -/- — — y- — .

^ \— x^2-\-x'^ l — x^'dx l+a;4

, 1, (^+1)2 1 ^ _i2^-l . ,. ,

3. y:=^l-^ — \ — 1— TTrtan. ^ — - —, m which

^^ (l+3aj + 3a;2)i ^y^ 1

X ' dx xt{\-\-xy

64. From the eqnatf
sin. X -\- sin. 2x +

. n-\-\ . 72
sm — x^m. ^x

+ sin. nx =z ?

sm. -a;

°"'-2
prove, by differentiating both members, that
cos. a; + 2 cos. 2ir + 3 cos. 3x -j- . . . + w cos. wa? is equal to
n+1 . 1 . 2/2+1 1 / . n4-l \2

sin. - 'i^sin '■

-t-1 . 1 . zrt+1 1 / . n+1 V

• 2I
sm. j^ a;

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DIFFERENTIAL CO-EFFICIENTS, 59

81D. X sm

(|+a,)sin.(2^ + a.) . . . 8m.(!^« + «)

sm. tna;

in which m is a positive whole number, prove that

cot. ic + cot.f — +a; )+ . . . cot.f n-\-x]=:mGoi.mx.

. \^ J \ ^ I

* As the equations assumed in this and the preceding example are not usually
given in treatises on elementary trigonometry, they will be demonstrated in the
Key to this work.

(

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SECTION IV.

SUCCESSIVE DIFFERENTIAL CO-EFFICIENTS,

4:8. The dififerential coeflScient of a function, /(x), of
a single variable, being in general another function, /'(oj),
of the same variable, we may subject this new function
to the rules by which f'{x) was derived from f{x)y and
thus obtain the second derivative, or differential co-efficient,
of the original function. The second differential co-efficient
will, in turn, give rise to a third, and so on ; and we thus
arrive at the successive derivatives, or differential co-efficients,
of a function.

The notation by which these successive differential co-effi-
cients are indicated will be best explained by an example ; —

Let us take y = a;" ; then

-^ = y^ = na?«-^ 1"* diffi co-efficient.

^1 z= y^' = n (n — 1) ic»-2 . . 2'* diff. co-efficient.

_y _. y{m) ^th jjg' co-efficient.

These are the first, second, . . . w*^ differential co-efficients
of the function y =:/(x). It is sometimes convenient to de-
note these by writing the function itself with as many dashes
as there have been difierentiations performed: thu3/'(aj),
f"(x)^ . . . f^"'\x)j are the first, second, . . . m*'* differen-

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DIFFERENTIAL CO-EFFICIENTS, Gi

tial co-efficients of /(a?), and have the same signification as

In the example just given, it is evident, that, if n be a
positive integer, the m^** differential co-efficient will be inde-
pendent of rr, that is, a constant, when m = n ; and that the
function will not have a differential co-efficient of a higher
order than the vP^* In other cases and forms of function,
there will be no limit to the number of difierentiations that
may be performed.

The symbols J^> ^.> ■ ■ ' ^' ^^-

. , , , dx dx^ dx"*-^ J

equivalent to —^ — f — -^ — j • • • -— 7 — j ^^^

(XX ax Q/X

are read second, third, . . . m*^ differential co-efficient of y
regarded as a function of x; and are to be viewed as wholes,
and not as fractions, having d^y, d^y, . . . d"^y, for their
numerator, and dic^, dx^^ . . . cfo;"*, for their denominators:
nor must the indices 2, 3, ... m, be considered as exponents
of powers, but as denoting the number of times the function
has been differentiated.

4:9. Successive difierential co-efficients of the product of
two functions of the same variable. Leibnitz' Theorem.

Take u = y«, in which y and z are functions of x ; then, by

Art. 20, we have

du dz dy ^

Tx^ydi^^lbc'
and, diflFerentiating both members of this equation with respect
to X, we have

d'^u __ d'^z dy dz dy dz d'^y

_^ d^z dy dz d'^y

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C2 DIFFERENTIAL CALCULUS.

In like manner, we should find

dx' ^^^ dx''^ dx dx'' '^ dx dx' '^^dx''
and

dx' -^y dx''^ dx dx' "^ dx' dx' '^ dx dx^'^^ rfx* *
This has been carried far enough to enable us to discover
inferentially the laws which govern the numerical co-efiicients,
and the indices of differentiation in the expressions for

d'^ XL d^ XL d XL

, , — -. These laws are the same as those for the

dx^ dx^ dx^

co-efficients and exponents in the Binomial Formula; for, in

respect to differentiation, y may be regarded as y^^^, and

z as 2(«\

To prove these laws to be general, let us assume them

to hold when n is the index of differentiation. Then

dx^ "~ ^dx» ^ dx dx^'-' ^ 2 dx' dx"-'^ ' ' '
(71—1) (n— 2). ._^(72^-^H-l) d'y d'^'^z

+ ^ 273rr.^^^ dx"" d^^

(n — l)(n-2) . . . {n^r) d^^^y d^-^'-^^'^z

+ n

2.3. . . r(/-+l) dx"-^^ dx''-^''-^^"^

4- . . .4-2 — ^.
^ ^ dx'^

Differentiating both members of this equation with respect

to x, reducing, and arranging the result, we find

{n + l) n. . . {n + l-r)d-+hjd—^z A^J'^^^y

"^ 273. . . (r + 1) dx^^^ dx^-"- "I" • ' ' t" dx^+''

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DIFFERENTIAL CO-EFFICIEXTS. C3

Now, the laws of the co-efficients and indices in this devel-
opment are the same as those assumed to be true in that
from which it was immediately derived ; but, by actual opera-
tion, we know them to hold when n = 4 : they therefore hold
when n 1= 5 ; and so on : hence they are universal.

As an example of the above, take u = e^^y ; then, observing

that-^ = a^'e*^, we find

ax"

Now, by examining the expression within the parentheses,
we discover, that if f a -| - p J y be developed by the Binomial

Theorem, treating the symbol -r- as a quantity, and [ ^~]y
/d\K /dy , ,, , J , dn d'^1/ d^'y

i^dx)^: • • W^ ^' ''^'^ '^^^'^'"^ ^^dx^-dx^ • • • -dx^^

d" u
we get that factor of .the development of -i— - : hence

d"u _d''{e'^y) _ ^/ , dy
d^ d^ -^ y^'^'dx)^

is a convenient and abridged form of writing the n'^ dififeren-
tial co-efficient of the function u = e^y,

SO. If n be a positive whole number, we may prove that

^ dJo'^ "" lix^ ^ 'dx^^ \' dxj^ 1. 2 ~ dx^-^ \^ dx\

_&c. + &c. . . . +(-1)m/£^. . . (1).

For let y=zuVj in which both u and v are functions of x;
then, differentiating with respect to a:, we have

dy d. uv do du

dx~^ dx ^ dx dx

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T

rz d'y d*y

: _" 1-: Tif to discover
LZL i T. yjl c«>efficients,
:_f expressions for

L'l Ls :1 -se for the

. I F-nnula: for, in
'^-: i as y ^\ and

:: -5 assume them
-::n. Then

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J

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