Derivatives. 165 



2. I" such an expression as /(^.r, j/^ the two quantities x and 

 V are entirely unrestricted in value and independent of each 

 other ; but if we have an equation like f(x, jKJ = o, then x and jy 

 are to some extent restricted ; any value may indeed be given to 

 one of the quantities but then the equation fixes the value of the 

 o^Ae?\ or in other words, either one of the quantities x or y de- 

 pends upon the other one; e. g., \if(x,y) stands for x—y-\-2 

 then w^hen this is not put eaual to anything there is no relation 

 between .v and y. We may let .^=3 and y=^ or 7 or 10 or any 

 other number. But if we put this same function equal to zero, 

 the7i there is some relation between x and y and they are to soffie 

 extent restricted in value. We may let a'=3, but then j'=5 and 

 nothing but 5. 



3. If the equation F(x, y) = o can be solved for r, we can ex- 

 press y in terms of x, or y can be determined as a function of x. 

 If we thus determirue r we hayey=/(x). 



In this equation, y=/(x), we may look upon x as a variable, 

 and of course if x varies i' will also var>'. We may consider x to 

 vary in any way we please, but then the equation determines the 

 way in which r varies. For the reason just stated x is called the 

 independent variable, and r, which is a function oi x, is called the 

 depeyideni variable. 



4. In the equation y=f(x) if a value be given to x then r will 

 have some corresponding value, and if x be given another value 

 different from the first one then y will have some value different 

 from the one it had at first. Moreover, the amount by which y 

 thus changes in value will depend in some way upon the amount 

 by which x changes, or in other words, there is some relation 

 connecting the change in value of y with the change in value of 

 X. This relation we will examine, and it will be found to be a 

 ver}' important relation in all that follows. 



5. Suppose f(x) to stand for 2x-\-\, then putting this equal to 

 y we have 



