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Find the second order derivatives of the functions given in Exercises 1 to 10. e ^ x sin 5x

Q6  Find the second order derivatives of the functions given in Exercises 1 to 10. 

      e ^x \sin5 x

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Given function is
y= e^x\sin 5x
Now, differentiation w.r.t. x
\frac{dy}{dx}=e^x.\sin 5x +e^x.5\cos 5x = e^x(\sin5x+5\cos5x)
Now, second order derivative is
\frac{d^2y}{dx^2}= e^x(\sin5x+5\cos5x)+e^x(5\cos5x+5.(-5\sin5x))
         = e^x(\sin5x+5\cos5x)+e^x(5\cos5x-25\sin5x)=e^x(10\cos5x-24\sin5x)
                                                                                                               =2e^x(5\cos5x-12\sin5x)
Therefore,  second order derivative is \frac{dy}{dx}=2e^x(5\cos5x-12\sin5x)

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