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Question

Find the second order derivatives of e6xcos3x

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Solution

Let y=e6xcos3x
dydx=ddx(e6x.cos3x)=cos3x.ddx(e6x)+e6x.ddx(cos3x)
=cos3x.e6xddx(6x)+e6x.(sin3x).ddx(3x)
=6e6xcos3x3e6xsin3x ....(1)
d2xdy2=ddx(6e6xcos3x3e6xsin3x)=6ddx(e6xcos3x)3ddx(e6xsin3x)
=6[6e6xcos3x3e6xsin3x]3[sin3xddx(e6x)+e6xddx(sin3x)
=36e6xcos3x18e6xsin3x3[sin3x.e6x.6+e6xcos3x.3]
=27e6xcos3x36e6xsin3x=9e6x(3cos3x4sin3x)

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