## A Treatise on Infinitesimal Calculus: Differential calculus. 1857 |

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Page 32

Dividing both sides by dx, we have

last member is of the form 9 f but will be a finite quantity, if our presumption is

correct. Let us represent it by fix), so that .-.

from ...

Dividing both sides by dx, we have

**dy _**d.fjx) _ fjx + dx)-fjx) dx dx dx of which thelast member is of the form 9 f but will be a finite quantity, if our presumption is

correct. Let us represent it by fix), so that .-.

**dy**= d.fx) = f\x)dx: (5) and thereforefrom ...

Page 42

... x*}* {a2 + (x + *a>)*}i &.x{a2 + x2}b — x2 ax {a2-f a?2}-* {tf + x2}* {dt + ix + ox)*}

* (a24a^) {a2+(# + A#)2}* whence, taking differentials, and omitting dx, where it is

added to the finite quantity x, we have , a%dx

... x*}* {a2 + (x + *a>)*}i &.x{a2 + x2}b — x2 ax {a2-f a?2}-* {tf + x2}* {dt + ix + ox)*}

* (a24a^) {a2+(# + A#)2}* whence, taking differentials, and omitting dx, where it is

added to the finite quantity x, we have , a%dx

**dy**a2 V ~ {a2 + x2}^' '' {a2 + a?}$ Page 54

<l>(x) x/Qr) •□ ay~ {*(»)}» omitting the infinitesimal <ir in the denominator,

because it is added to the finite quantity x ; and therefore . /'(*) <P(x)dx- <t>'(x)f(x)

dx

<l>(x) x/Qr) •□ ay~ {*(»)}» omitting the infinitesimal <ir in the denominator,

because it is added to the finite quantity x ; and therefore . /'(*) <P(x)dx- <t>'(x)f(x)

dx

**dy**= w • (5) Ex. 1.**dy**= a + bx b + ax"1 bdx(b + ax) — adx(a + bx) (b + ax)' dx. Page 55

Bartholomew Price. Therefore to differentiate x" we have the following rule :

Multiply by the exponent, diminish the exponent by unity, and multiply by dx.

Suppose the exponent to be negative, then y = x~n,

suppose n ...

Bartholomew Price. Therefore to differentiate x" we have the following rule :

Multiply by the exponent, diminish the exponent by unity, and multiply by dx.

Suppose the exponent to be negative, then y = x~n,

**dy**— —nx-("+Vdx. Andsuppose n ...

Page 58

tfWf{x) dx. (11) 34.] Differentiation of loga x. Let y = loga y + Ay = loga (x + ax); Ay

= logaOr + AaO-logatf, but when — becomes an infinitesimal, viz. — , loga(l + — )

X X * X ' dx by Cor. II, Lemma I, Art. 21 ; hence log, a

tfWf{x) dx. (11) 34.] Differentiation of loga x. Let y = loga y + Ay = loga (x + ax); Ay

= logaOr + AaO-logatf, but when — becomes an infinitesimal, viz. — , loga(l + — )

X X * X ' dx by Cor. II, Lemma I, Art. 21 ; hence log, a

**dy**= ; .-. d.logaX — - .### What people are saying - Write a review

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algebraic curves algebraical angle asymptote axis becomes Calculus change of sign changes sign circle coefficients conic constant coordinates corresponding critical values curvature cycloid decreases determine direction double point drawn dy _ elimination ellipse epitrochoid equa equal equicrescent equivalent explicit function expression factors finite quantity geometrical given point Hence homogeneous homogeneous function hyperbola hypocycloid hypotrochoid imaginary increases infinitesimal Infinitesimal Calculus infinity involving l)th let us suppose logarithmic maxima and minima maximum or minimum minimum value negative normal nth degree number of points observed ordinate parabola parallel partial derived-functions pass perpendicular plane curve plane of reference point of inflexion points of intersection polar positive properties radius real roots right-hand member roots of f(x shewn Similarly singular value straight line substituting symbol tangent Theorem tion Tractory triangle vanish whence Witch of Agnesi zero

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