Quotient Rule f(x)=(x+1)/(x-1) Hey everybody, welcome back. We’re going to keep going with quotient rule problems. This one is a little bit different. It is f(x)=(x+1)/(x-1). Let’s go ahead and write out the quotient rule so that we have it here for reference. It is d(x), f(x)/g(x)=f1(x), g(x)-f(x), g1(x)/g(x)2. And all these means is when you’re trying to take the derivative, that’s what d(x) stands for of a function with f(x) on the top and g(x) on the bottom. So like in our problem, f(x) here is one, g(x) is x+1. You use this formula to take the derivative. So this problem is slightly different because we have to apply quotient rule. First to this is 1/x+1, and then to this, 1/x-1. So, let’s just go ahead and go through this. The derivative of course we call f1(x) equals. So we’re just going to look at 1/x+1 first. So, let’s do that. And remember, one is f(x) here because they’re on the top and x+1 is g(x) because they’re on the bottom. So we’re going to apply the quotient rule to this, 1/x+1. So first thing is f1(x), which means the derivative of f(x), the top here is one. So the derivative of one being constant is zero. So we have zero, and then let’s go ahead and say g(x) which is just x+1. So 0*x+1, and then we say –f(x) which is one, the top here, times g1(x) or on the derivative of g(x), which is the derivative of the bottom which is just one. So, we did the top and now, we need to divide by the bottom squared, which is x+1 and then squared. So we’ve got to had it already and applied the quotient rule to this fraction here. So now we need to apply it to this fraction. So we’ve done that. We say minus, which I’m cropping from here, and then go through it again. So the derivative of the top, one is zero times, we need the bottom and top, x-1 minus the top, f(x) which is one, times the derivative of the bottom, g(x) and the derivative of that is one. The derivative of x-1 is one. And then we divide by the bottom squared with so that’s x-1, so times x-12. Okay, so we have applied quotient rule. First we want to write x-1, and then 1/x-1. This is for x+1, the 1/x+1 and this is the part for 1/x-1, and we’re subtracting this one from this because it tells us to subtract here. So we’ve got our whole formula written now. And now all we need to do is simplify this. So, let’s go ahead and simplify. Zero of course gets rid of this whole thing because it’s multiplied by zero, same here and whole thing is going to cancel, which is multiplied by zero. So, 1*1 of course is just one so we’re going to end up with a -1 on top over here, and then we can say over and x+12. You could multiply it our but this is simpler, just to leave it this way. And then over here, we’ll put the minus sign from here. And then we have 1*1=1 so this is a -1 on top here. So we could say -1/x-12. So the only thing that we want to do is we have minus and then a negative. So those are going to cancel each other out and become a positive. So I’m going to go ahead and erase that one and make this positive. And then the only other thing I’d like to do is actually flip this because if you can, you don’t usually like to lead the entire function with a negative sign, we’ve got a negative here. So I’m going to flip this so that we got the positive in front so we don’t have to leave with the negative. So this is actually just going to become – I’m going to write this one first so it’s going to be 1/x-12, and then we’ll subtract because we have the negative sign here, minus 1/x+12. So this is as simple as we’re going to get here. We applied the quotient rule to our function of course for this fraction, and then to this one, simplified and this right here is going to be our final answer.