82 PRACTICAL MATHEMATICS 



dx 1 dy 1 dy dx 



Then j- = j- j = ~T> or j^ x j- = 1 



dy dy dx dx dx ay 



dx dy 



52. In general, if the law of the curve is ?/ =/(#), 

 then y+ By = f(x + Bx) 



By=f(x+Bx)-f(x) 



By f( x+ Bx)-f(x) 

 Bx Bx 



and then -r- is the limiting value of the fraction - J- - 

 dx 8x 



when Bx is made infinitely small. 



This gives us a general method of determining the slope of a 

 curve when the law of the curve is known, and this process of 

 finding the slope is called differentiation. 



In order to avoid going through this process each time we 

 wish to work with a certain curve, we establish the results of 

 differentiating well-known functions of x and use these as standard 

 results. 



y = ax n where a and n are constants. 

 If y = ax n 



then y + By = a(x + Bx) n 





When Bx is made infinitely small, all of the terms on the right- 

 hand side involving Sx and powers of Bx can be neglected 



du 



and -~ = anx n ~ l 



dx 



It should be noticed here that a and n, the constants of the 

 curve, are constants of different types, a is a constant multiplier, 

 and remains a multiplier during differentiation ; n is a constant 

 power and differentiation diminishes it by unity, while the result 

 is multiplied by n. 



We can apply this result to differentiate any function of x 

 of the form y = ax n . 



