 In this video, we provide the solution to question number five from practice exam number three for math 1210 We're asked to compute the derivative of e to the x times the natural log of x You'll notice that there is a product of two functions going on here So we need to use the product rule So this is going to look like e to the x prime times the natural log of x plus e to the x times the natural log of x prime So we take the derivative of these things separately the derivative of e to the x is itself So we get e to the x times the natural log of x even though the natural exponential in the natural log are Inverses of each other when you multiply them together the inverse function property does not apply that only applies when we compose them So there's no simplification. We can see there yet for the next one We're gonna get e to the x then we get the derivative of the natural log Which was one over x and so let's look to see what we can find I don't see that as an exact answer But notice if you factor out the e to the x you'll get e to the x times 1 over x plus the natural log of x So the sums in a different order and they factor the e to the x but is equivalent to the derivative We just computed so the correct answer is f