 In this video, we provide the solution to question number nine for the practice final exam for math 1210 We're asked to find the derivative of the function f of x equals the square root of x times e to the x We want to calculate the derivative and I see we have a product of two functions the square root of x times e to the x So the derivative will be calculated using the product rule So we have to take the derivative of the square root of x times e to the x And then we're going to take the square root of x times the derivative of e to the x The derivative of the square root of x that's one that you might just have memorized You've done it so often that the square root the derivative of the square root of x is going to be 1 over 2 times the square root of x Then you get e to the x if you're not sure where that came from Just remember that the square root of x is the same thing as x to the one half power So if we calculate the derivatives of these things using the power rule We're going to get one half times x to the negative one half power negative exponents mean it goes in the denominator and One half exponents means you have the square roots. That's where the 1 over 2 times the square root of x is coming from So continuing on we get 1 over 2 times the square root of x times e to the x That's in the numerator e to the x and then we've got the square root of x times the derivative e to the x Which that is itself e to the x and so now let's look for the result that Resembles what we have right here You don't see exactly what I have in front of us But it looks like a lot of the answers have e to the x factored out We could try doing that if you factor out e to the x You're gonna end up with 1 over 2 times the square root of x plus the square root of x like so For which then that really looks like choice a if we factor out the square root of x from the denominator notice if you do that if you factor out the One excuse me if you factor out the square root of x from the denominator You're gonna be left behind with a one half plus in this case an x which of course if you divide it by the square Of x that's the same thing as having e to the x times x the negative one half power times one half plus x So sometimes when we calculate our derivative, we might have done a little bit differently than what could have been expected on a question There's nothing wrong with that. They're completely valid methods So we have to be able to check which for a multiple choice question That is which of these results is equivalent to the one we computed in which case we see that choice a is equivalent to the derivative of this function