 In this video we provide the solution to question number 23 for the practice final exam for math 12-10 We're given the function f of x equals x squared plus x plus 1 and we're asked to set up and Simplify the difference quotient f of x plus h minus f of x over h And so to do that we're just going to start off by writing down the thing that we need to set up and simplify f of x plus h minus f of x all over h Right here in which case f of x plus h means we're going to put an x plus h in for each of the x's in this formula So we're going to get an x plus h squared plus x plus h plus 1 And then we're going to subtract from that f of x which we can just copy down the formula as it's given That is to say x squared plus x plus 1 and this all sits above the h right here So now we need to foil expand some things before we can combine some like terms in particular this x plus h squared We need to foil that thing out upon doing so we end up with x squared plus 2x h plus h squared Then we'll just drop the parentheses everywhere else x plus h plus 1 They're not necessary and then we have this x squared plus x plus 1 all above the h For which case then there should be everything in this last group should cancel with something in the first group So the x-square is cancel the x cancels and then the plus 1 cancels right there if we look at what does not cancel out We have a 2x h We have an x an h squared Excuse me, and we have an h and so if we've done everything correctly everything in the numerator should now be divisible h We're going to factor that h out. We will give us 2x plus h plus 1 all over h So then the divisor of h in the numerator cancels of the divisor of h in the denominator And now our difference quotient is simplified We get 2x plus h plus 1 and this is where we stop it notice the instructions did not ask us to compute a derivative We're not currently taking the limit as h goes to infinity But if we did if we set h equal to 0 you get 2x plus 1 as you know is the derivative of the function x squared plus x plus 1