Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.
$ f(t) = 2.5t^2 + 6t $
Okay here we have a function. F F T. I'm going to use capital T. So as I'm doing all this work, my lower case T. S don't confuse you and look like plus times. I'm going to use capital T. So F F T. Is defined to be 2.5 times T squared plus six times T. The domain of this function. The domain of a function ah is in this case is the variables T. The domain would be all the values of T where this function is defined. Well, this function is defined for any number that you plug into T. So the domain is, you know, all the real numbers, the entire X axis. The next thing we want to do is we want to calculate uh F prime of T. The derivative of F with respect to T. Using uh the definition of derivative. So the definition of the derivative for F prime of T is going to be to limit of F of T plus delta T minus F of T. All divided by delta T. And we take the limit of this expression as delta T approaches zero. So F of T plus delta T. Wherever you see T. In the original function, we're gonna uh plug in T plus delta T. So F F T plus Delta T. is going to be 2.5. And instead of times T squared OK, we are doing T plus delta T to the second Plus six times T. But remember right now we're writing the expression for F of T plus delta T. Uh So instead of tea, we're going to be writing in T. Plus delta T. So plus six times this variable is really plus six times to T. Plus delta T. So that takes care of the F. Of T. Plus delta T. Now we have to subtract F. Of T. Which is the 2.5 T square plus 60. So we're going to subtract 2.5 T sq. And since we're subtracting FFT and this entire expressionist FFT, we're subtracting each of these terms. So I already wrote minus 2.5 times T squared. Now, since once again, since I'm subtracting FFT, I also had to subtract 2 60. So that's why we write -16. Now all of this gets divided by delta T. And I forgot to write the little limit. Um but so we'll just squeeze it in here. We want to take the limit of this whole expression as delta T approaches zero. So what I'm going to do next is I'm going to do T plus delta T to the second. Using algebra. If you think of foil basically this is a binomial. Um multiplying a binomial, you would really have T plus delta T. To the second is T. Plus delta T. Times itself. That's what it means to raise it to the second. So T plus delta T. Times T. Plus delta T. You can use foil to multiply two by no mules. And I'm gonna start changing the color just to help make it look a little less cluttered here. So we have 2.5 and that has two times two T. Plus delta T squared. Well, T plus delta T square T. Plus delta T. Times itself is going to be T squared plus two times T times delta T. Plus delta T squared. So all of that in princes has to be multiplied by 2.5. Uh And then we have to do, I'm just going to bring this down for now and then I'm going to scroll down on the screen and will continue simplifying this expression. At this point. It looks really confusing, but a lot of things will drop out nicely. So plus six times T plus delta T minus 2.5 T. Times T squared minus 16. And all of that has to be divided by delta T. So let's scroll down so that what we have in blue is near the top of this page so that we have some room to continue simplifying. And let's change the green for now. So we want to take the limit of this entire expression. This delta T approaches zero. Remember what we're trying to do? We're calculating the derivative of the given function. Using the definition of the rib there. It's a 2.5 has to be distributed as it multiplies each to three terms and Princes. So we're going to have 2.5 T squared plus 2.5 times to T delta T. 2.5 times two is five. So plus five times T delta take Then the 2.5 is going to talk to delta T squared plus 2.5 delta T squared. And then plus Now six has to multiply these two terms. So plus 60 plus six times delta T. Ah And then we're minus seeing 22.5 T squared. And we're minus ng or subtracting 2 16. And all of this once again is over the delta take. All right, let's move down a little bit more and bring what we have in green near the top. All right. This is where things will start simplifying a little bit because we'll be able to combine some terms. So, the limit of this entire expression is delta T approaches zero. Well, let's combine like terms. First of all. 2.5 T squared minus 2.5 T squared cancels. So that works out nice. Um we have plus 60. subtract 60 that cancels out lace. We have a five T Delta T. We have 2.5 delta T squared. And we have a six delta T. So, these are all unlike terms. So we're gonna write all three of these five, she delta T Plus 2.5 delta T square. Uh and then plus six Delta T. And all of that needs to be divided by delta. Take Now each of these three terms in the numerator are being divided by delta T. So we can separate ah these additions in the numerator and take each one of these three different terms and individually divided by delta T. So five tee times delta T. Over delta T. The delta teeth would cancel out. You would have five T. Here 2.5. Uh let's write down the plus time here. The 2.5 times delta T squared divided by delta T. Or delta T squared divided by delta T. Leads you just a delta T. Up here's a 2.5 delta T. And then here we have a plus and then six delta T divided by delta T. Two. Delta teas will cancel and you'll have just six. So now as we take the limit of this expression five T plus 2.5 times delta T. Plus six as delta T. Goes to zero. The delta T. Here is going towards zero out of these three terms. Uh This middle term to 2.5 delta T. Is the only term that has adult it T. So is the delta T approaches zero. Here's a delta T. It's approaching zero. So 2.5 uh times delta T. Is going to approach zero because if delta T. Is approaching 2.5 times delta T approaches zero. This five T. Is going to stay five T. Okay because it's delta T. That approaches zero, not the T. So five T stays five T. Likewise two plus six days plus six. So when we take this limit, when we take the limit of this expression, five T plus 2.5 times delta T plus six, as delta T approaches zero, we end up with five T Plus six. So that is the derivative of the function FF. T. Okay, so this just moved down a little bit. Yeah, this is our answer for F prime of T. And we evaluated it or we found it by using the definition, the limit definition of the derivative. So at prime of T equals five T plus six. Uh If we went to the top of the problem and we found a derivative, the old fashioned way, it should be five T plus six. All right. Uh If we found a derivative of this 2.5 times T squared would be two times 2.5 uh times T. Which would be five T plus derivative 60 is six. So derivative of this is five T plus six. And that's exactly what we found basically doing it. More complicated way using the definition of the derivative, the limit definition of the derivative last but not least. What is the domain of the F prime of T. The derivative function. Well, the derivative is F I'm sorry, the derivative is five times T plus six. This is defined for any real value of T. So basically, T could be any number. So the domain of F prime A. T once again is the entire X axis, any number on the X axis or basically any real number.