Derivatives. 175 



19. Examples. 



Find the derivative with respect to .v of the following ex- 

 pressions : 



/. 2-v^-f 4Ji-^+jf. 4.. x^ -\- 2^x -\- 2 . 



2. -r^+,r^H-.r+i. 5. x^—x-\-\. 



J. x^—\. 6. x^-\-\. 



20. I'o FIND THE Derivative with Respect to x of the 

 Product of two Functions of x. 



Let 21 and v be the two functions of x, and let r be their prod- 

 uct, then we have 



y^uv. (i) 



Now increase x by Jx and suppose the corresponding amounts 

 by which y, n, and z' respectively increase are Jr, -in, and Jv, 

 then we have 



Expanding (2) we get 



y-\- Jy=uv-\-?iJv-\-vJii-\- JuJz', f^) 



or y + Jy=uv+uJv-^(v-\-Jv)Jn. (^j 



Subtract (i) from (^) and we get 



Jy=uJv+(v+Av)Mi. C5; 



Divide both sides oi (^) by Jx and we get 



Taking the limit of each side of (6), remembering that the last 

 term of the right-hand member is the product of two quantities, 

 hence its limit equals the product of their separate limits, and 

 that M' approaches zero as J.v approaches zero, hence the limit of 

 7'-(- J7'=7', we get 



limit f J|'l .. . Jz' , ,. . J// 



or ' T>,y=uTi_,.v-\-vT>jc. (S) 



21. To FIND THE Derivative of the Product of any 

 Number of Functions of x. 



First, take three functions of .v, say w, z', and w, and let y be 

 their product, then we have 



y^=^iivu'. (i) 



