224- MISCELLANEOUS EXAMPLES. 



- tan" 1 x 

 9. Evaluate when x = 1. 



log ( 



cot 



10. Evaluate , when x = 0. 



cot x + log x 



sec" x , TT 



11. Evaluate -^^- when # = ^- 



tan nx tan mx 



12. Evaluate = n - ^-r- , 



sin (n x mx) 



(1) when a; = 0, (2) whenn = m, 



13. In the equation f(x + h)f(x)=hf'(x+6h), shew 



that if /"(or) is not zero the limiting value of as h is 



indefinitely diminished is - : also shew that if f r (x) 



is the first of the differential coefficients ^"(2;), /'"(#),... 

 which is not zero, the limiting value of 6 as h is in- 

 definitely diminished is 



14. In the equation f(x + Ji) f(x) =hf'(x + Oh) shew 



that if be the same for all values of h, it must equal 



- and/" (a:) must be constant. 

 2 



15. Change the independent variable from z to x in the 



equation 



j<fy dy n Na ( 2 cPy , dy\ 



* j^ + z -j- - 1 = Gg Z Y \ z TI + a j r 

 df da { d z dz} 7 



where =e sin;r . 



