) y y Hence ( – x ) y – xy = EXERCISE . Find the second order derivatives of the functions given in Exercises to . . x + x + .
x log x . e x sin x . e x cos x . tan – x .
log (log x ) . sin (log x ) . If y = cos x – sin x , prove that . If y = cos – x , Find in terms of y alone.
. If y = cos (log x ) + sin (log x ), show that x y + xy + y = . If y = A e mx + B e nx , show that m n mny . If y = e x + e – x , show that .
If e y ( x + ) = , show that = . If y = (tan – x ) , show that ( x + ) y + x ( x + ) y = Miscellaneous Examples Example Differentiate w.r.t. x , the following function: (i) (ii) log (log x ) Solution (i) Let y = ( ) ( ) Note that this function is defined at all real numbers x > − . Therefore ( ) ( ) ( ) ( ) −− + − = − This is defined for all real numbers x > − .
(ii) Let y = log (log x ) = log (log ) log7 x (by change of base formula). The function is defined for all real numbers x > . Therefore (log (log )) log7 (log ) log7 log x dx log7 log Example Differentiate the following w.r.t. x .