Examples of the Processes of the Differential and Integral Calculus |
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Page 1
... dy dx dy dx ' y being some function of x , and u some function of y . This theorem may be extended to any number of functions , so that du du dv ds dy = dx da dv ds dy Ex . ( 1 ) Let u = ( a + bx " ) " . Then y = a + bx " , u = y TM , dy dx ...
... dy dx dy dx ' y being some function of x , and u some function of y . This theorem may be extended to any number of functions , so that du du dv ds dy = dx da dv ds dy Ex . ( 1 ) Let u = ( a + bx " ) " . Then y = a + bx " , u = y TM , dy dx ...
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... dy then - ( - ) dx = log y -y log If sin ya sin ( a + y ) , dy da = sin ( a + y ) cos y x cos ( a + y ) If y " log y = ax , dy dx ( 46 ) If tan y = dy dx ( 47 ) Let tan = y 2 = -1 a y - 1 ( 1 + nlogy ) 1 + x sin y , ( cos y ) 2 sin y ...
... dy then - ( - ) dx = log y -y log If sin ya sin ( a + y ) , dy da = sin ( a + y ) cos y x cos ( a + y ) If y " log y = ax , dy dx ( 46 ) If tan y = dy dx ( 47 ) Let tan = y 2 = -1 a y - 1 ( 1 + nlogy ) 1 + x sin y , ( cos y ) 2 sin y ...
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... dy { a2 − 2 ( x2 + y2 ) } x dx { b2 + 2 ( x2 + y2 ) } y ' = - ( 52 ) Let ( a + y ) 2 ( b2 — y2 ) — x2 y2 then dy dx = y2 ( b * — y3 ) } y3 + ab2 = 0 , Functions of Two or more Variables . ( 53 ) u = ( - ) ( 54 ) du dx = du = u = du ...
... dy { a2 − 2 ( x2 + y2 ) } x dx { b2 + 2 ( x2 + y2 ) } y ' = - ( 52 ) Let ( a + y ) 2 ( b2 — y2 ) — x2 y2 then dy dx = y2 ( b * — y3 ) } y3 + ab2 = 0 , Functions of Two or more Variables . ( 53 ) u = ( - ) ( 54 ) du dx = du = u = du ...
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... dy'dx drs u = dx'dy s = 1 , u = xy " ; r = 1 , du dx mxm - 1y " ; = d2 u dy da = m n xm m - 1 1y " - 1 du dy = = n - 1 nx " yr- d'u dx dy ( 2 ) W = x2 + y2 x2 du dy dx = - 1 ' = 1 , s = 1 , y2 8xy a2 + y2 ( x2 - y2 ) " r = 1 , 8 = 1 , ď u 1 ...
... dy'dx drs u = dx'dy s = 1 , u = xy " ; r = 1 , du dx mxm - 1y " ; = d2 u dy da = m n xm m - 1 1y " - 1 du dy = = n - 1 nx " yr- d'u dx dy ( 2 ) W = x2 + y2 x2 du dy dx = - 1 ' = 1 , s = 1 , y2 8xy a2 + y2 ( x2 - y2 ) " r = 1 , 8 = 1 , ď u 1 ...
Page 23
... dy dx = 30 y x2 - y2 ; ( y2 + x2 ) 2 r = 1 , 8 = 1 ; d u = dx dy u = x sin y + y sin x ; r = 1 , 8 = 1 ; ( 8 ) d2 u d'u = cos y + cos x = dy da dx dy ( 9 ) u = sin x cos y ; r = 2 , 8 = 2 ... dx dy dz = == = SUCCESSIVE DIFFERENTIATION . 23.
... dy dx = 30 y x2 - y2 ; ( y2 + x2 ) 2 r = 1 , 8 = 1 ; d u = dx dy u = x sin y + y sin x ; r = 1 , 8 = 1 ; ( 8 ) d2 u d'u = cos y + cos x = dy da dx dy ( 9 ) u = sin x cos y ; r = 2 , 8 = 2 ... dx dy dz = == = SUCCESSIVE DIFFERENTIATION . 23.
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