THE RULES FOR DIFFERENTIATION 87 



57. The Hyperbolic Functions. 

 (1) If y = sinh x 



then y = - (<* - e~*) 



dy 



- (e* + e~*) = cosh x 



(2) If y = cosh a; 



then y = - (e* + e~*) 



= - 

 dx 2 V 



= sinh x. 

 58. The Rules for Differentiation. 



(a) Now ~ = x -KT- since is a fraction, and the value of a 

 8x ox Sz ox 



fraction remains the same when numerator and denominator are 

 multiplied by the same quantity. Let 8z represent the increase 

 in some function z, which itself depends upon x, then when Bx 

 is made infinitely small &z becomes infinitely small. 



Then ^ = ^ x- 



1 11C11 ~ - r> A ., 



OX OZ OX 



Now j is the slope of a chord of the curve obtained by plotting z 



OZ 



DfJ 



horizontally and y vertically, and this becomes -^, the actual slope 



0A 



of the curve when &r, and therefore Sz, become infinitely small. 



02 d% 



Similarly, when Bx is made infinitely small - becomes -j-, the 



Ox dx 



actual slope of the curve obtained by plotting x horizontally 

 and z vertically. 

 Thus, in the limit, when Sx is made infinitely small, 



dy _ dy dz 



dx dz dx 



The following examples will illustrate the use of this rule : 

 (1) To differentiate sin n x. 



Then y = z n where z = sin x 



dy . dz 



-j- = nz n ~ l and -j- = cos x 



dz dx 



