96 PRACTICAL MATHEMATICS 



Example 2. Differentiate e sin bx. 



du dv du 

 (a) Using the rule ^ = <^+<^ 



u = e x and -^ = ae 

 ax 



v = sin bx and -r- = b cos to 



Then = be a cos & + ae" sin bx 



ax 



= e (b cos te + a sin &r 



= d sin &#(& cot & + a) 

 I dy I du I dv 



- 



Xhen - = a+bcotbx 



y ax 



and ^ = ^ sin &(a + 6 cot 



aa; 



(c) Working logarithmically 



X + log e Sin & 



b cos 5# 



y ~dx " sin fe 



= a + b cot bx 



^ = e sin fca: (a + b cot 

 ax 



Example 3. Differentiate 

 (a) Working logarithmically 

 = log (1 



1 dy _ 2a? _ 1 f -2a? \ 

 w dx 1 + x 2 2^1 x z j 



2x 



~ 



2x(l - x*) + x(l + a; 2 ) 



wC* -r 2 ^ 



it-^O U/ ^ 



