DIFFERENTIAL AND INTEGRAL CALCULUS. 57 



r^r~5ui (a-x) */(2ax-a?), fjq~ =^7. i 



\ 9 



"Za- 



cosa: 



5. Explain the notation -~^ -..., and find the w th 



WiC tfiC 



differential coefficients of sin a?, cos a;, log a?, and sin 3 a?. If 

 y = cot" 1 (a;), prove that 



-j- n =(!)" I n 1 sin"?/ sinwy. 

 If y =a ^ logx, S=> 



6. Express , ' in terms of ?/, 0, and their several 



differential coefficients. Find the n th differential coefficients 

 of x 3 sin#, (1 - x*) cot" 1 (a?) ; and prove that the w th dif- 

 ferential coefficient of x cosfcc is 



(a 2 + b^ z x cos (bx 4- n tan' 1 f-)| . 



7. If 

 then will 



and ^ 



DIFFERENTIAL CALCULUS. 

 EXPANSIONS OF FUNCTIONS IN SERIES. 



Taylor's Theorem. 



If we assume that a function </> (x+-h) can be expanded 

 in a series of ascending integral powers of A, and assume 



72 rn 



that expansion to be A Q + AJi + A^r- H-...+ A n -. - 4... , 



L_ L^ 



where y! , A^ ... -4 n do not involve 7< at all, i.e. are func- 



I 



