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Question

7。e® cos 3x

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Solution

The given function is e 6x cos3x.

Let y= e 6x cos3x.

Differentiate function y with respect to x.

dy dx = d( e 6x cos3x ) dx dy dx = d( e 6x ) dx .cos3x+ d( cos3x ) dx . e 6x dy dx =6 e 6x ( cos3x )sin3x( 3 e 6x ) dy dx =3 e 6x ( 2cos3xsin3x )

Again, differentiate with respect to x.

d dx ( dy dx )= d( 3 e 6x ( 2cos3xsin3x ) ) dx d 2 y d x 2 =3( d( e 6x ) dx ( 2cos3xsin3x )+ d( 2cos3xsin3x ) dx . e 6x ) d 2 y d x 2 =3( 6 e 6x ( 2cos3xsin3x )3( 2sin3x+cos3x ). e 6x ) d 2 y d x 2 =3( 12 e 6x cos3x6 e 6x sin3x6 e 6x sin3x3 e 6x cos3x )

Simplify further,

d 2 y d x 2 =3( 9 e 6x cos3x12 e 6x sin3x ) d 2 y d x 2 =9 e 6x ( 3cos3x4sin3x )

Thus, the second order derivative of the given function is 9 e 6x ( 3cos3x4sin3x ).


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