CHAPTER XIII. 



DERIVATIVES. 



I. Notation. A definition of a function of a quantity was 

 given in I, Art. i. To designate a function of x we use the 

 notation /j^jij. 



A function of a quantity is denoted by writing the quantity in 

 a parenthesis and. writing the letter f ox F or some other func- 

 tional symbol before the parenthesis, e. g. 



f(x), F(x), F/x) denote functions of .v, 



f(y), F(y),fjy) denote functions of j', 



f(x-\-h), F(x-\-h),f'(x + /i) denote functions of^'H--^, 



f(a), F(a),f„(a) denote functions of ^. 



The student must be careful not to look upon the expression 

 f(x) as meaning /times x. The symbol /as used here is not a 

 quantity at all, but simply an abbreviation for the words 

 jtindion of. 



It frequently happens that in the same discussion we wish to 

 refer to different functions of x, in which case we use different 

 functional symbols, as F( x ) , f( x ) , fj x ) , /„( x ) , F„(x), etc. 



It also frequently happens that in the same discussion we wish 

 to refer to the same function of different quantities, in which case 

 we use the same functional symbol before the parenthesis but dif- 

 ferent quantities within the parenthesis, e. g. If f(x) denotes 

 x^-j-i then f(a) denotes a^-^i,f(2) denotes ,r+i, etc., and if 

 F(x) denotes \/x-\-2, then F(y) denotes Vj'+3, F(x-\-/i) de- 

 notes \/ X -{-7l -{- T„ ^tC. 



A function of two quantities is any expression in which both of 

 the quantities appear. 



If we have to deal with a function of two quantities, say x and 

 y, we use the notation /f'.^-, y) or F(x, y), and if, in the same dis- 

 cussion, we wish to speak of two or more functions of x and y, 

 different functional symbols are used, 3isf(x\y),f,(x,y), etc. 



