DIFFERENTIAL AND INTEGRAL CALCULUS. 5 



such an equation is used the members are called diffe- 

 rentials of y, and of $> (x) respectively. 



Another most useful way of considering a differential 

 coefficient is by drawing the curve y = $(x), which can, 

 always be done, or conceived to be done ; for whatever 

 the relation between x and y be, we can by measuring 

 off a sufficient number of values of x along a fixed straight 

 line, and all the corresponding values of y at right angles 

 to them, obtain as many points as* shall give a-clear notion 

 of the form of the curve. 



Suppose then OM=x, MP= $ (x), MN=fi (fig. 3) ; 

 therefore ON=x+h, NQ (f)(x-{-h)j and draw PR perpen- 

 dicular to NQ, then RQ = $(x+Ji)-<$>(x), or if QP meet the 



C TT x T>TT <f> (X + fl) <f) (X) AT 



axis of x in c/, tan PUx = j . Now when 



Q moves up to P, the limiting position of the straight 

 line QPU is the tangent PT to the curve at P, or 



rtrr r .^ <f> (x -}- h) 4> (x) dy ,. . 



tanPJb = limit - = or 6 (x}. 



h ax 



Of course if we solved the equation for x and obtained 

 it in the form x = *fy(y), the curve represented by this 

 equation would be the same as before, since all the values 

 which satisfy one must satisfy the other, and since we 



should then have exactly as above tanPT'^/ = -^- , we 

 d>i dx 



^TxTf 1 - 



We will now proceed to find the differential coefficients 

 of simple functions of a?. 



(1) x\ n being constant, i.e. not changing as x changes. 

 Taking y = x\ we have 



y + by = (x + Ax) n or (x + h) n , or Ay = (x + h) n - x\ 



and - = - 1+- -1 



Ace h h (\ x) 



Now the expansion of (1+2)" by the Binomial Theorem 

 is arithmetically true whenever z is numerically less than 



