92 PRACTICAL MATHEMATICS 



Then if z log e y, -*- is required, 



dz dz du 



but T- = j- x ~r 



ax ay ax 



dz 1 



and _ = _ 



ay y 



Hence ^=1^ 



cte y ax 



That is, - -j- is the result of differentiating log^t/ with respect 



c/ 



to x. 



To differentiate # n e sin n a?. 



? = x n e^ sin" a; 



log e ?/ = n logeX + nx+ n log e sin x 



TV.CC * * l dy (I cosafl 



Differentiating --^ = wi-+l+ -. h 



sin 



= -{ 1 + x + x cot x} 

 ar 



Multiplying throughout by y or x n e sin" x 



-j- = nx n ~ l e m sin n x{l + x + x cot x} 

 dx 



We can use logarithmic differentiation to establish the dif- 

 ferential coefficients of cot (ax + b), sec (ax + b), and cosec (ax + b), 

 and these results can be used as standard forms. 



cos (ax + b) 



(I) y = cot (ax + b = -: -. rf 



sin (ax + b) 



Then \og e y = log e cos (ax + b) - \og e sin (ax + 6) 



1 du a sin (ax + b) a cos (ax + b) 



Differentiating - -f- = ; rr : ; rt 



y dx cos (ax + b) sin (ax + b) 



/sin 2 (ax + b) + cos 2 (ax + 5)1 

 X sin (ax + b) cos (ax + b) J 



a 

 sin (ax + b) cos (ax + b} 



dy _ a cos (ax + 5) 



dx ~ sin (cw? + b) cos (o# + 6) sin (ax + b) 



a 



sin 2 (ax + b) 

 = a cosec 2 (ax + b) 



y = sec (CKC + b) = 



cos (o# + b) 



cos 



