CHAPTER IV. 



DIFFERENTIAL COEFFICIENTS OF THE INVERSE TRIGONOME- 

 TRICAL FUNCTIONS AND OF COMPLEX FUNCTIONS. 



58. LET y ^> (x), so that y is a known function of x ; it 

 follows from this that x must be some function of y, although 

 we may not be able to express that function in any simple 

 form. The best mode for the reader to convince himself of 

 this will be to recur to algebraical geometry and suppose x 

 and y to be the co-ordinates of a point in a curve the equation 

 to which is y = <j>(x). For every value of x there will be 

 generally one or more values of y, positive or negative, as 

 the case may be. So for any value of y there will be 

 generally one or more definite values of x, which, as they 

 really exist, may be made the subjects of our investigations, 

 even although our present powers of mathematical expres- 

 sion may not always furnish us with simple modes of repre- 

 senting them. 



59. A simple example will be given in the equation 



y = x*-2x+l .. (1). 



Solve this equation with respect to x, and we have 



= iy* (2). 



Here (2) shews that if any value be assigned to y we must 

 have for x one of two definite values. 



Now in (1), x being the independent variable and y the 

 dependent variable, we have by Arts. 28, 33, and 44, 



^-2a-2 



7 ftJb ~~ a 



dx 





