 Hi and welcome to the session. I am Shashi and I am going to help you to solve the following question. Question is, find the second order derivatives of the functions. The given function is e raised to the power x, vertical i pi sin 5x. Let us start the solution now. First recall, let us assume y is equal to e raised to the power x sin 5x. Now differentiating both sides with respect to x, we get dy upon dx is equal to e raised to the power x multiplied by derivative of sin 5x plus sin 5x multiplied by derivative of e raised to the power x. Now this implies dy by dx is equal to e raised to the power x multiplied by cos 5x multiplied by 5 plus e raised to the power x multiplied by sin 5x. As we know derivative of sin 5x is equal to cos 5x multiplied by 5. Here we have applied the chain rule. Now and the derivative of e raised to the power x is equal to e raised to the power x. So dy by dx is equal to 5 e raised to the power x cos 5x plus e raised to the power x sin 5x. Now again differentiating both sides with respect to x, we get d square by upon dx square is equal to 5 multiplied by e raised to the power x multiplied by minus 5 sin 5x cos 5x multiplied by e raised to the power x. Here we have applied the product rule plus again applying the product rule for this function we get e raised to the power x 5 cos 5x plus sin 5x multiplied by e raised to the power x. Now we get second order derivative of y is equal to minus 25 e raised to the power x sin 5x 5 e raised to the power x cos 5x plus 5 e raised to the power x cos 5x plus e raised to the power x sin 5x which is equal to 5 e raised to the power x cos 5x plus 5 e raised to the power x cos 5x is equal to 10 e raised to the power x cos 5x. Again these two are height terms so we get minus 24 e raised to the power x sin 5x. Now let us take e raised to the power x common and 2 is also common so 2 e raised to the power x is common and we get 5 cos 5x minus 12 sin 5x. So the required second order derivative of y is given by d square by upon dx square equal to 2 e raised to the power x multiplied by 5 cos 5x minus 12 sin 5x. So our required answer is 2 e raised to the power x multiplied by 5 cos 5x minus 12 sin 5x. This completes the session. Hope you enjoyed the session.