7。e® cos 3x
The given function is e 6 x cos 3 x .
Let y = e 6 x cos 3 x .
Differentiate function y with respect to x .
d y d x = d ( e 6 x cos 3 x ) d x d y d x = d ( e 6 x ) d x . cos 3 x + d ( cos 3 x ) d x . e 6 x d y d x = 6 e 6 x ( c o s 3 x ) − sin 3 x ( 3 e 6 x ) d y d x = 3 e 6 x ( 2 c o s 3 x − sin 3 x )
Again, differentiate with respect to x .
d d x ( d y d x ) = d ( 3 e 6 x ( 2 c o s 3 x − sin 3 x ) ) d x d 2 y d x 2 = 3 ( d ( e 6 x ) d x ( 2 cos 3 x − sin 3 x ) + d ( 2 c o s 3 x − sin 3 x ) d x . e 6 x ) d 2 y d x 2 = 3 ( 6 e 6 x ( 2 cos 3 x − sin 3 x ) − 3 ( 2 sin 3 x + cos 3 x ) . e 6 x ) d 2 y d x 2 = 3 ( 12 e 6 x cos 3 x − 6 e 6 x sin 3 x − 6 e 6 x sin 3 x − 3 e 6 x cos 3 x )
Simplify further,
d 2 y d x 2 = 3 ( 9 e 6 x cos 3 x − 12 e 6 x sin 3 x ) d 2 y d x 2 = 9 e 6 x ( 3 cos 3 x − 4 sin 3 x )
Thus, the second order derivative of the given function is 9 e 6 x ( 3 cos 3 x − 4 sin 3 x ) .
The given function is
Let
Differentiate function
Again, differentiate with respect to
Simplify further,
Thus, the second order derivative of the given function is
