DIFFERENTIAL AND INTEGRAL CALCULUS. 16 



rential coefficients obtained severally by considering all 

 the functions but one to be constant, All the above rules 

 are included in this. This may be expressed as follows : if 

 y = F(u, v, iV) ...), where u, v, w are functions of a?, 



dy fdF\ du fdF\ dv 



~ = -j ) -j- + - T - -.=- +..., the brackets signifying 



dx \dujdx \dvjdx & / "8 



fJTp\ 



partial differentiation, viz. that in forming f -= ) all the 



functions v, w, ... except u are regarded as constant. 

 Thus, to differentiate or"*, (1) considering the upper ic's 

 to be constant, we get of x x* x ~\ (2) considering the centre 

 x as the only variable, and the differential coefficient is 

 x** logic.ic.ic*"" 1 or of of* logic, (3) considering the upper x 

 as the only variable, and we have x* x logic (x* logic) or 



ofof* (loga?) 2 , whence the true value of -~ is 



x x x xfc j- + log ie+ (logx)' 



(X 



This may be tested by taking the logarithmic differential 

 coefficient of both members of the equation y y? x . 



QUESTIONS ON THE PRECEDING. 



1. Define the terms function, independent variable, and 

 explain what is meant by the limit of a function which for 

 a particular value of the variable assumes an unmeaning 

 form. Prove that the limit of secic tanic, as x approaches 

 the value -|TT, is ; and that of sec 2 icsecic tanic is J. 



2. Give exa'mples of functions which are limited in 

 either sign or magnitude although the independent variable 

 on which they depend is capable of all values from co to 

 co. State how -any of the following functions are limited, 

 x being capable of all values : 



1 i / 1 \ x x z 



since, tanic, ic a + , (1 -f x}* , f 1 4- - ) , -^ 



* \ SO/ 00 



