1820.] of Taylor's Theorem, S^c. 103 



Then will log. y = I y = x I a, and d {I y) = ~ = dxla. 



Therefore 

 dy = y . d X . I a 



d* y = d y . d X . I a =. y . d x"^ {1 aY 

 d^ y = d y . d x^ {lay = y . d x^ (la)" ■ 

 d* y = dy .dx^ilaY =y . d x* {I a)* 

 &c 



Then A J, = A («') = <,y + ^ + ^ + ^ + &c. ^ 

 Or A («') =y.dx.la + ^-^'J^ + ll.'fSiJ!}l + 5cc. 



= a" (^A o; . / a + j + ,^ ^ + Sec. J. 



If fl = e, be the number whose logarithm is unity, then / a, 

 {I a)", (/ ay, Sec. are each equal to unity, we shall then have 



A (.0 = .' (a . + ^ + .^ + ,-^ +'&c.) = e' (e- - 1). 



Example 15. — To find the increment of sin. x (radius unity). 

 Suppose d X = A X = w, a constant quantity, and j/ = sin. x. 



Then c? y = d (sin.j;) = -{- w cos. j? = + A x cos.x 



d'^ y = d { w cos. x) = — w- sin. x =■ — A x^ sin. a; 



d^ y =z d ( — w'^ sin. x) = — w^ cos. x = — A x^ cos. x 



d* y = d (— w^ COS. x) = + w* sin. x = + A x* sin. ar 



d^ y = d ( w* sin. a;) = + w^ sin. x = + A x^ cos. a: 



rf» (/ rf3 y rf« y 



Sec. 



Therefore A .,y = A (sin. a) = d y + '-^ + "^l + ,^^ + 

 Ur A (sui. a:) = A a- cos. x — - sm. a- — - — - cos. a- + 



2 2.S '2.3.4 



sm. X + COS. a: — &c. 



Example 16. — To find the increment of cos. x (radius unity). 

 Put 7v = d X = A x, a constant quantity, andy = cos. x. 



Then d y z= d (cos. x) = — w sin. a: = — A a- sin. je 



d^ y = d {— tv sin. x) = — W^ cos. a: = — Ax* cos. x 



d^ y = d {— w"^ cos. x) = -\- lo^ sin. x =■ + A x^ sin. x 



d* y = d { w^ sin. x) = + tv^ cos.x = + Ax* cos. jc 



Sec 



Therefore A (cos. x) = dy + '^ + fl + ^^ + &c. 

 Ur A (cos. x) = — A X sm. x cos. x + ;r-x sm. x + 



A j:< „ 



cos. X — &C. 



2.3.4 



Erample 17.— To find the increment of tan. x (radius unity). 

 Put ?« = J X = A X, a constant quantity, and ?/ = tan. x. 



