TANGENT TO A CIRCLE. 25 



easily seen. In treating any example without the aid of 

 general rules, we should frequently find our success depen- 

 dent upon our readiness in effecting such transformations; but 

 the next two Chapters will explain methods of making the 

 problem of ascertaining any differential coefficient depend 

 upon the knowledge of those of a few standard functions. 



41. From analytical geometry we know that the equation 

 y = ^(a a # 2 ) represents a circle, and it is also known from 

 the principles of that subject that the tangent at the point 

 (x, y) of a circle is inclined to the axis of x at an angle 



OR 



whose trigonometrical tangent is r-r-^ . Also in the 



Y I CL ~~" CO } 



case of a circle the straight line which we have defined as the 

 tangent is the same straight line as that which fulfils the con- 

 dition of " touching the circle," given in Euclid, Book in. 



42. In the Chapters on the geometrical application of the 

 Differential Calculus we shall recur to the subject of tangents. 

 We have given the above example here that the student may 

 at this early period acquire the conviction that important uses 

 may be made of a differential coefficient. 



43. The,following is another geometrical application. The 

 area OAMP, see the diagram to Art. 35, must be some func- 

 tion of x, since it is a definite quantity when we assign a 

 definite value to x, and varies when x varies. Denote this 

 function by u, and PQ by Ao; ; then 



u + Aw = area OANQ, 

 therefore A?/ = area MNQP; 



therefore Aw lies between MP. PQ and NQ.PQ, 

 that is, between ykx and (y + Ay) Ace ; 



therefore -r- lies between y and y + Ay. 



Hence, diminishing A#, and therefore Ay, without limit, we 

 have 



du _ 



dx~ y ' 



