denoted by f ″ ( x ). It is also denoted by D y or y ″ or y if y = f ( x ). We remark that higher order derivatives may be defined similarly. Example Find , if y = x + tan x .
Solution Given that y = x + tan x . Then dx = x + sec x Therefore sec = x + sec x . sec x tan x = x + sec x tan x Example If y = A sin x + B cos x , then prove that . Solution We have dx = A cos x – B sin x and = d dx (A cos x – B sin x ) = – A sin x – B cos x = – y Hence + y = Example If y = e x + e x , prove that .
Solution Given that y = e x + e x . Then dx = e x + e x = ( e x + e x ) Therefore = e x + e x = ( e x + e x ) Hence + y = ( e x + e x ) – ( e x + e x ) + ( e x + e x ) = Example If y = sin – x , show that ( – x ) x dx . Solution We have y = sin – x . Then ( or ( So ( ) .
or ( ( dx dx or ( Hence ( x dx Alternatively , Given that y = sin – x , we have , i.e., ( So ( ) . (