 Hi and welcome to the session. I am Shashi and I am going to help you with the following question. Question is find the second order derivatives of the functions. The given function is e raised to the power 6x multiplied by cos 3x. Let us start the solution now. First of all, let us assume y is equal to e raised to the power 6x multiplied by cos 3x. Now differentiating both sides with respect to x we get dy upon dx is equal to where we will apply the product rule e raised to the power 6x multiplied by derivative of cos 3x plus cos 3x multiplied by derivative of e raised to the power 6x. This implies dy upon dx is equal to minus 3 e raised to the power 6x sin 3x plus 6 e raised to the power 6x cos 3x. We know derivative of cos 3x is equal to minus 3 sin 3x. So, and the derivative of e raised to the power 6x is equal to 6 multiplied by e raised to the power 6x. So, we get dy by dx is equal to minus 3 multiplied by e raised to the power 6x multiplied by sin 3x plus 6 multiplied by e raised to the power 6x multiplied by cos 3x. Now again we will differentiate on both sides with respect to x. Now differentiating both sides with respect to x we get d square by upon dx square is equal to minus 3 multiplied by e raised to the power 6x multiplied by derivative of sin 3x plus sin 3x multiplied by derivative of e raised to the power 6x plus 6 multiplied by e raised to the power 6x multiplied by derivative of cos 3x plus cos 3x multiplied by derivative of e raised to the power 6x. Now we can write d square by upon dx square is equal to minus 3 multiplied by e raised to the power 6x multiplied by 3 cos 3x e raised to the power 6x multiplied by sin 3x plus 6 multiplied by e raised to the power 6x minus 3 sin 3x plus 6 e raised to the power 6x multiplied by cos 3x. Now this is equal to minus 9 e raised to the power 6x cos 3x minus 18 e raised to the power 6x sin 3x minus 18 e raised to the power 6x sin 3x plus 36 e raised to the power 6x cos 3x this implies d square by upon dx square is equal to 27 e raised to the power 6x cos 3x minus 36 e raised to the power 6x sin 3x so this implies d square by upon dx square is equal to we can take 9 e raised to the power 6x common and get 3 cos 3x minus 4 sin 3x in the bracket. So our required second derivative is equal to 9 multiplied by e raised to the power 6x multiplied by 3 cos 3x minus 4 sin 3x. So this is our required second order derivative. This completes the session. Hope you understood the session. Take care and goodbye.