Solution Given that y = x3 + tan x Then
dy
dx  = 3x2 + sec2 x
Therefore
2
2
d y
dx
 =
(
)
2
2
3
sec
d
x
x
dx
+
= 6x + 2 sec x sec x tan x = 6x + 2 sec2 x tan x
Example 36 If y = A sin x + B cos x, then prove that 
2
2
0
d y
y
dx
+
= Solution We have
dy
dx  = A cos x – B sin x
and
2
2
d y
dx
 = d
dx  (A cos x – B sin x)
= – A sin x – B cos x = – y
Hence
2
2
d y
dx
 + y = 0
Example 37 If  y = 3e2x + 2e3x, prove that 
2
2
5
6
0
d y
dy
y
dx
dx
−
+
=