Examples of the Processes of the Differential and Integral Calculus |
From inside the book
Results 1-5 of 100
Page 1
... 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 ' 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 ...
Page 5
... dx ( x + 2 ) 3 ( ( x + 3 ) " ) x + 1 ( 37 ) u = x2 , log u = x log x , du ( 38 ) 6868 ( 39 ) ( 40 ) da = ( 1 + log x ) ... dy dx dx du dy ( 43 ) Let ( 44 ) ( 45 ) DIFFERENTIATION . 5.
... dx ( x + 2 ) 3 ( ( x + 3 ) " ) x + 1 ( 37 ) u = x2 , log u = x log x , du ( 38 ) 6868 ( 39 ) ( 40 ) da = ( 1 + log x ) ... dy dx dx du dy ( 43 ) Let ( 44 ) ( 45 ) DIFFERENTIATION . 5.
Page 6
... 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 = - x ( cos y ) 3 + x . ; taking the logarithmic differential we find ( 48 ) If ( 49 ) y dy dx dy dx = = = sin y 1 - x2 1 + x ...
... 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 = - x ( cos y ) 3 + x . ; taking the logarithmic differential we find ( 48 ) If ( 49 ) y dy dx dy dx = = = sin y 1 - x2 1 + x ...
Page 7
... dy " { y - x - 2 ( xy ) } yś dx + { x - y - 2 ( xy ) } x3 dy 2 ( xy ) 3 ( x + y ) 2 * dx 2x ( x + y ) 2 du = du du ( 55 ) u = = xy , = = Yxy - 1 , = x2 log x , dx dy du = xy 20 ( 56 ) u = log du dx = du = da + log ad · ( x + ( x2 - y3 ) ...
... dy " { y - x - 2 ( xy ) } yś dx + { x - y - 2 ( xy ) } x3 dy 2 ( xy ) 3 ( x + y ) 2 * dx 2x ( x + y ) 2 du = du du ( 55 ) u = = xy , = = Yxy - 1 , = x2 log x , dx dy du = xy 20 ( 56 ) u = log du dx = du = da + log ad · ( x + ( x2 - y3 ) ...
Page 22
... 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 ...
Other editions - View all
Common terms and phrases
a² b2 a²x² angle arbitrary constant assume asymptote becomes branches C₁ Cambridge circle co-ordinates condition Crelle's Journal curvature curve cycloid determine differential coefficients differential equation dx dx dx dy dx dx² dy dx dy dy dy dy dz dz dz eliminate ellipse equal Euler factor formula fraction function Geometry gives Hence hypocycloid infinite intersection John Bernoulli Let the equation lines of curvature locus logarithmic logarithmic spiral Multiply negative origin parabola perpendicular radius SECT singular points singular solution spiral Substituting subtangent surface tangent plane theorem triangle University of Cambridge vanish whence x²)³