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# Justifying the power rule

## Video transcript

what I want to do in this video is to see whether the power rule is giving us results that at least seem reasonable this is by no means a proof of the power rule but at least it'll it'll will feel a little bit more comfortable using it so let's say that f of X f of X is equal to X the power rule tells us that F prime of X is going to be equal to what well X is the same thing as X to the first power so n is implicitly 1 right over here so we bring the 1 out front it'll be 1 times X to the 1 minus 1 power so it's going to be 1 times X to the 0 power x to the 0 is just 1 so it's just going to be equal to 1 now does that make conceptual sense if we actually try to visualize these functions so let's let me draw let me actually try to graph these functions so that's my y-axis this is my x-axis and let me graph y equals x so y is equal to f of X here so Y is equal to X so it looks something like that so Y is equal to X so this is f of X equals 2x or Y is equal to this f of X right over there now the derivative actually let me just call that f of X just to make it not to confuse you so this right over here is f of X is equal to X that I graphed right over here Y is equal to f of X which is equal to X and now let me graph the derivative let me graph F prime of X that's saying it's one that's saying it's 1 for all X regardless of what X is it's going to be equal to 1 is this consistent with what we know about derivatives and slopes and all the rest well let's look at our function what is the slope of the line or the tangent line right at this point well the slope of this right over here this has slope 1 continuously or it has a constant slope of 1 slope is equal to 1 no matter what X is it's a line the law and for line the slope is constant over here the slope is indeed 1 if you go to this point over here the slope is indeed 1 if you go over here the slope is indeed 1 so we got a pretty we got a pretty valid response there now let's try something where the slope might change so let's say I have G of X is equal to X square the power rule tells us that G prime of X would be equal to what well n is equal to 2 so it's going to be 2 times X to the 2 minus 1 or it's going to be equal to 2 X to the first power it's going to be equal to 2x so let's see if this gives us that this makes reasonable sense and I'm going to try to graph this one a little bit more precisely graph it a little bit more precisely so let's see see how well I can graph it precisely I can graph it so this is x-axis y-axis let me mark some stuff off here so this is 1 2 3 4 5 this is 1 2 3 4 1 2 3 4 so G of X G of X when X is 0 its 0 when X is 1 it is 1 when X is negative 1 it's 1 when X is 2 it is 4 so that puts this right over there 1 2 3 4 let's just write over there when X is negative 2 you get 2 4 it's a parabola you've seen this for many years so it looks something like it looks something I put that point a little bit too high it looks something like this it looks something actually I put the last two points I graphed a little bit weird so this might be right over here so it looks something like this it looks something like that and then when you come over here it looks something like it looks something like that it's symmetric so I'm trying my best to draw it reasonably so there you go that's the graph of G of X G of X is equal to x squared now let's graph G prime of X or what the power rule is telling us that G prime of X is so G prime of X is equal to 2x so that's just a line goes through the origin of slope 2 so it looks something like that when X is equal to 1 Y is equal to 2 when X is equal to 2 y or G of X is equal to 4 so it looks something something like this something let me try my best to draw a straight line it looks something like this now does this make does this make sense well if you just eyeball it really fast if you look at this point right over here and you want to think about the slope of the tangent line the slope I'm going to try my best to draw let me do this in a color that's more that pops out a little bit more so the slope of the tangent line would look something like this so it looks like it has a reasonably high negative slope if I were to yeah it's definitely a negative slope and it's a pretty steep negative slope well for X for X is equal to two G prime of two G prime or sorry for X is equal to negative two G prime of negative two G prime of negative two is equal to two times negative two two times negative two which is equal to negative four so this is claiming that the slope at this point so this right over here is negative four saying that the slope of this point is negative four M is equal to negative four that looks about right it's a very it's a fairly steep negative slope now what happens if you go right over here when X is equal to zero well our derivative if you say G prime of zero is telling us that the slope of our original function G at X is equal to zero is two times zero is zero well does that make sense well if we go to our original parabola it does indeed look set it makes sense that's the slope of the tangent line the tangent line looks something like this we're at a minimum point we're at the vertex the slope really does look to be zero and what if you go what if you go right over here to x equals two the slope of the tangent line well over here the tangent line looks something like this looks like a fairly steep positive slope what is our derivative telling us based on the power rule it's saying G prime so this is essentially saying hey tell me what the slope of the tangent line for G is when X is equal to two well we figured it out it's going to be 2 times X it's going to be 2 times 2 which is equal to 4 it's telling us that the slope over here is 4 that their slope and now I'm just using mm is often the letter used to denote slope they're saying that the slope of the tangent line there is 4 which seems completely completely reasonable so I encourage you to try this out yourself I encourage you to to try to estimate the slopes by calculating by taking points closer and closer around those points and you'll see that the power rule really give you results that actually make sense
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