A Treatise on Infinitesimal Calculus: Differential calculus. 1857University Press, 1857 - Calculus |
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Page 42
... equa- tion , and omitting do when added to the finite quantity x , we have dy = ex dx { ex + 1 } 2 * If therefore ex ex f ( x ) = ex + 1 ' f ' ( x ) = { ex + 1 } 2 Ex . 4. To differentiate cos x sin 2x . 42 [ 24 . ILLUSTRATIVE EXAMPLES ...
... equa- tion , and omitting do when added to the finite quantity x , we have dy = ex dx { ex + 1 } 2 * If therefore ex ex f ( x ) = ex + 1 ' f ' ( x ) = { ex + 1 } 2 Ex . 4. To differentiate cos x sin 2x . 42 [ 24 . ILLUSTRATIVE EXAMPLES ...
Page 104
... equa- tion , we have 2x√1 = log ( 1 + √1 tan x ) - log ( 1 - √ - 1 tan x ) = √1 tan x + ( tan x ) 2 2 ( tan x ) 3 — -1 3 ( tan x ) 4 4 ( tan x ) 5 + -1 + ...... 5 - { - ~ √1 tan x + ( tan x ) 2 ( tan x ) 3 + -1 2 3 ( tan x ) ...
... equa- tion , we have 2x√1 = log ( 1 + √1 tan x ) - log ( 1 - √ - 1 tan x ) = √1 tan x + ( tan x ) 2 2 ( tan x ) 3 — -1 3 ( tan x ) 4 4 ( tan x ) 5 + -1 + ...... 5 - { - ~ √1 tan x + ( tan x ) 2 ( tan x ) 3 + -1 2 3 ( tan x ) ...
Page 116
... equa- tions will be formed ; y = f ( x ) , y + dy = f ( x + dx ) , y + 2dy + d2y = f ( x + 2 dx + d2x ) , y + 3dy + 3 d2y + d3y = f ( x + 3dx +3 d2x + d3x ) , · y + ndy + n ( n - 1 ) 1.2 d2y + n ( n - 1 ) ( n - 2 ) 1.2.3 d3y + ...... n ( ...
... equa- tions will be formed ; y = f ( x ) , y + dy = f ( x + dx ) , y + 2dy + d2y = f ( x + 2 dx + d2x ) , y + 3dy + 3 d2y + d3y = f ( x + 3dx +3 d2x + d3x ) , · y + ndy + n ( n - 1 ) 1.2 d2y + n ( n - 1 ) ( n - 2 ) 1.2.3 d3y + ...... n ( ...
Page 124
... equa- tions : the old and the new variables being equicrescent or not , as the case may be . All these several processes involve transformations of differ- ential expressions , and because such expressions commonly in- volve second and ...
... equa- tions : the old and the new variables being equicrescent or not , as the case may be . All these several processes involve transformations of differ- ential expressions , and because such expressions commonly in- volve second and ...
Page 128
... equa- tion , will be included under it . Let , as in Art . 49 , the general form of the function be where x , y , z , ... u = F ( x , y , z , . ) , ... ( 90 ) are independent of each other ; and are there- fore such that a variation of ...
... equa- tion , will be included under it . Let , as in Art . 49 , the general form of the function be where x , y , z , ... u = F ( x , y , z , . ) , ... ( 90 ) are independent of each other ; and are there- fore such that a variation of ...
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a₁ algebraical angles b₁ becomes Calculus change of sign changes sign circle coefficients constant cosec curve d2F d2F d²u d²x d²y d³u d³y derived function determine differential equation dr dr dr dy dx dx dx dy dx² dy dx dy dy dy dz dy² dz dx equal equicrescent explicit function expression F(xo F(xo+h factor finite quantity fraction func given Hence homogeneous function increases increments indeterminate form infinite infinitesimal Infinitesimal Calculus infinity involved logarithms loge Maclaurin's maxima and minima maximum or minimum minimum value negative partial derived-functions positive primitive equation radius replaced result right-hand member roots Similarly sin x singular value Sturm's Theorem substituting suppose symbols Theorem tion vanish variables variation versin whence zero