Quotient Rule f(x)=(x-1)/(x2+2x+1) Hi everyone, welcome back, quotient rule problems. The first thing we’re going to do tonight is the following, f(x) =(x-1)/(x2+2x+1), and we’re going to be using quotient rule. F1(x)g(x)-f(x)g1(x)/g(x)2. So now you can barely read that. All this is saying is that when you are trying to take the derivative of function and you’ve got something on the top and something on the bottom. In this case, f(x) here is x-1 and g(x) is x2+2x+1, you follow this formula to take the derivative. So we’ll do the derivative of the top times the bottom minus the top times the derivative of the bottom divided by the bottom squared. So, that’s all that it’s saying. So we’re just going to go ahead and apply the quotient rule here. So, let’s go ahead and do that and we’ll say f1(x) equals first thing, derivative of f(x) which is the top. So the derivative of x-1, which is just one, and then times the bottom. So x2+2x+1 minus the top times the derivative of the bottom. So the derivative here, the derivative of the first term x2 is 2x, and the derivative of the second term is two so plus two and the one goes away because it’s a constant. So that’s the derivative of the bottom. So that’s the entire top of the equation, and then we divide by g(x) or the bottom squared. So we just write it out, x+2 xs+1 and we square that whole thing because the squared here is part of the quotient rule -- equation that we’re using. So, once we’ve taken our terms and apply them to quotient rule here, all we need to do is simplify. So the bottom is simplify as good as we’re going to get it. So, let’s go ahead and simplify the top and hopefully not too many steps. So x2, I’m just going to distribute the one. So that’s x2+2 xs+1, and then minus -- let’s put some parenthesis. So we’ll multiply this also. X*2x=2x2, and then x*2=2x, and then 1*2x, and then 1*2, so minus two. And I’m not going to rewrite the bottom because we don’t need to simplify it. So I’m just going to go for the simplifying the top here. So, that’s going to look like -- let’s see. x2+2x+1, and I’m going to go ahead and distribute the negative sign here. So I have -2x2-2x+, you got a negative times a negative, so that’s a +2x, and then again +2. And let’s see. We should be up to simplify that. x2-2x2 is going to be a negative x2, and we’ve got 2x-2x=0+2x=2x+2x, and then 1+2=3. So we’ve got everything. So remember, this was just asks simplifying the top so I’m going to go ahead and get up some more room here by raising this guy, or answers actually going to be f1(x)=, and remember this was the top so x2+2x+3, and then we said we had simplified the bottom as well as we could already. So the bottom is just going to be x2+2x+12. That’s the answer. So we took the function and we applied quotient rule to find the derivative. We got this, and then we went through a set of simple case and steps and eventually we’re on to put this answer. So that’s how you use the quotient rule to take the derivative of a function like this.