 In this video we provide the solution to question number three for practice exam number three for math 1210 We're asked to compute the second derivative of e to the x times cosine of x So let's say that this function here for the sake of it is just a y So we have to calculate the second derivative that means we have to first calculate the first derivative as the name suggests So we have to take the derivative here of e to the x times cosine of x this will require the product rule So we have to take the derivative of e to the x times cosine and Then we have to add to that e to the x times the derivative of cosine For which then we see that the derivative e to the x is Equal to e to the x we get a cosine of x and then we get the derivative of cosine, which is a negative sign So we then get the derivatives e to the x cosine of x minus e to the x sine of x Now I want you to watch out here That's gonna be tempted because choice number a is the first derivative This is equal to y prime. That is a distractor. We need to look for the second derivative Now in order to calculate the second derivative, we can just compute it the way it is or if you want to you could perhaps factor it Do whatever you want in order to make the next derivative calculation a little bit easier if you think it might be difficult So if we compute the second derivative this time We see that the second derivative you're gonna have to use the product rule You're gonna take the derivative e to the x which is itself then you times that by cosine x minus sine of x And then you have to add to that e to the x times the derivative of cosine minus sine Which derivative cosine is a negative sign and then the derivative of negative sign is gonna be a negative cosine Like so for which it looks like there might be some like terms that could be put together here It's like if we factor out the e to the x We're gonna have a cosine x minus sine x You then get a minus sine x and then a minus cosine x so notice that the cosines actually cancel out we're gonna end up with a negative 2 e to the x sine of x and so that then leads us to choose option e Which is the correct second derivative