Examples of the Processes of the Differential and Integral Calculus |
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Page 2
... power of the logarithm of a , but the nth logarithm of that quantity , 1 du dx a log a log2 x ...... log - 1 x du ( 8 ) u = log ( sin x ) ; = cot x . dx $ 1 - Cos m x du m ( 9 ) U = = log ( ; 1+ cos m x dx sin ma du 2 ( 10 ) u = log ...
... power of the logarithm of a , but the nth logarithm of that quantity , 1 du dx a log a log2 x ...... log - 1 x du ( 8 ) u = log ( sin x ) ; = cot x . dx $ 1 - Cos m x du m ( 9 ) U = = log ( ; 1+ cos m x dx sin ma du 2 ( 10 ) u = log ...
Page 4
... powers , it is generally most convenient to take the differential of the logarithm , or , as it is usually called , the logarithmic differential of the function . ( 31 ) Let u = ( a + x ) " ( b + x ) " , ( 32 ) ( 33 ) ( 34 ) ( 35 ) log ...
... powers , it is generally most convenient to take the differential of the logarithm , or , as it is usually called , the logarithmic differential of the function . ( 31 ) Let u = ( a + x ) " ( b + x ) " , ( 32 ) ( 33 ) ( 34 ) ( 35 ) log ...
Page 9
D. F. Gregory. CHAPTER II . SUCCESSIVE DIFFERENTIATION . THE analogy between Algebraic powers and successive differentials , when expressed by the notation of Leibnitz , was observed soon after the invention of the Calculus . Leibnitz ...
D. F. Gregory. CHAPTER II . SUCCESSIVE DIFFERENTIATION . THE analogy between Algebraic powers and successive differentials , when expressed by the notation of Leibnitz , was observed soon after the invention of the Calculus . Leibnitz ...
Page 45
... and denominator by e " , then +1 y = - whence e2 = y + 1 and 2x = log y + 1 ' y - 1 y - 1 and differentiating , dy = 1 − y2 . ( 12 ) Eliminate the power from the equation m dx ELIMINATION OF CONSTANTS AND FUNCTIONS . 45 9.
... and denominator by e " , then +1 y = - whence e2 = y + 1 and 2x = log y + 1 ' y - 1 y - 1 and differentiating , dy = 1 − y2 . ( 12 ) Eliminate the power from the equation m dx ELIMINATION OF CONSTANTS AND FUNCTIONS . 45 9.
Page 46
D. F. Gregory. ( 12 ) Eliminate the power from the equation m y = ( a + x2 ) " . Taking the logarithmic differential we have dy dx = 2 m ay na2 + x2 ( 13 ) Eliminate the functions from y = sin ( log x ) ; d'y dy the result is + + y = 0 ...
D. F. Gregory. ( 12 ) Eliminate the power from the equation m y = ( a + x2 ) " . Taking the logarithmic differential we have dy dx = 2 m ay na2 + x2 ( 13 ) Eliminate the functions from y = sin ( log x ) ; d'y dy the result is + + y = 0 ...
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