LOGARITHMIC DIFFERENTIATION 91 



u+ Sw u 



and 8y 



- 5 --- 



V+ QV V 



uv + v $u uv u 



v(v + 8i>) 

 v 8w u 8u 



Making &r infinitely small, 



, 

 ai/ 



dx 

 To differentiate 



v -j -- 

 dx ax 



on. , 



Then u = log^ and -7- = - 



/tC CD 



. dv 



v = x* and -r- = 2x 

 dx 



dx 



du dv 



V -; -- t^^- 



dx dx 



-- 2x 

 x 



X* 

 12 \OgeX 



~~lx*~ 



59. Logarithmic Differentiation. A method which can be used 

 to great advantage is that of taking logarithms of both sides 

 and then differentiating throughout. 



For example, let y = 



Then lg2/ = log e w + b log e u c log e o> 



As u, v, and w are functions of x, the right-hand side can be 

 easily differentiated, as we have a simple means of differentiating 

 the logarithm of a function of x (see 58 (a) ) ; but we must also 

 be able to differentiate log e y with respect to x. 



