tailieunhanh - Handbook of mathematics for engineers and scienteists part 42

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 42', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Differential Calculus for Functions of a Single Variable 255 . Methods for interpreting other indeterminate expressions. 1 . Expressions of the form 0 and - can be reduced to indeterminate expressions or by means of algebraic transformations for instance 0o transformation rule 0 œ 0 0 0 z z z z u x u x v x I . . i 1 1 1 . . . u x - v x ------------- ----------- transformation rule u x v x f u x v x 2 . Indeterminate expressions of the form 1 0 00 can be reduced to expressions of the form or by taking logarithm and using the formulas ln uv v ln u - . Example 2. Let us calculate the limit lim cos xf x . x 0 We have the indeterminate expression 1 . We find that ln lim cos lim ln cos lim lim X x 0 x 0 x 0 X x 0 x2 Therefore lim cosx 1 x e-1 2 . x 0 -yj e . Higher-Order Derivatives and Differentials. -tan x 1 lim ------- -- x 0 2x 2 Taylor s Formula . Derivatives and differentials of higher orders. The second-order derivative or the second derivative of a function y f x is the derivative d y of the derivative f x . The second derivative is denoted by y and also by yXx f x . The derivative of the second derivative of a function y f x is called the third-order derivative yw y . The nth-order derivative of the function y f x is defined as the derivative of its n - 1 th derivative . y ra-1 . The nth-order derivative is also denoted by yXn pn f n x . The second-order differential is the differential of the first-order differential d 2y d dy . If x is the independent variable then d2y y dx 2. In a similar way one defines differentials of higher orders. . Table of higher-order derivatives of some elementary functions. x n a a - 1 . a - n 1 xa-n In x n -1 n-1 n - 1 -1 xn sin x n sin x 2 sinh x n _ i cosh x if n is dd sn x I sinh x if n is even ax n In a nax - x _1 n-12 cos x n cos x n cosh x n - f cosh x if n is even I sinh x if n is odd. 256 Limits and Derivatives . Rules for calculating higher-order derivatives. 1. Derivative of a sum .