52 DIFFERENTIAL AND INTEGRAL CALCULUS. 



and if we assume 



f 



11 1 } 



logo; + - + r +...+ \ , 



1 n n- I nr\ 1 



the same rule. 



d* n y I n 

 If we put r n in [A] we get at once -7-^ = 5 



whence all subsequent ones are known. For purposes 

 explained in the next section, it is often convenient to 

 obtain a rational equation connecting two or more dif- 

 ferential coefficients of any function. Suppose, for instance, 

 y = sin" 1 ^, therefore 



'--- 



the required equation. Differentiating this n times, we have 



or 



So if y = (sin-'-o^, J(\ - x z ) = 2 suT 1 *, 



x 2 



or (1 - x 2 ) -~ - x -- = 2, which of course leads to the same 



-~ - -- 



