If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:7:58

so I've talked about the partial derivative and how you compute it how you interpret it in terms of graphs but what I'd like to do here is give its formal definition so it's the kind of thing just to remind you that applies to a function that has a multi variable input so X Y and you know put I'll just emphasize that it could actually be you know a number of other inputs you could have 100 inputs or something like that and as with a lot of things here I think it's helpful to take a look at the one dimensional analogy and think about how we define the derivative just the ordinary derivative when you have a function that's just one variable you know this would be just something simple f of X and you know if you're thinking in the back of your mind that it's a function like f of x equals x squared and the way to think about the definition of this is to just actually spell out how we interpret this DF and DX and then slowly start to tighten it up into a formalization so you might be thinking of the graph of this function you know maybe it's some some kind of curve and when you think of evaluating it at some point you know let's say you're evaluating it at a point a you're imagining DX here as representing a slight nudge just a slight nudge in the input value so this is in the X Direction got your x coordinate your output f of X is what the y-axis represents here and then you're thinking of DF as being the resulting nudge here the resulting change to the function so when we formalize this we're going to be thinking of a fraction that's going to represent DX so DF over DX and I'll leave myself some room you can probably anticipate why if you know where this is going and well instead of saying DX I'll say H so instead of thinking you know DX is that tiny nege you'll think H and I'm not sure why H is used necessarily but just having some kind of variable that you think of as getting small maybe all the other letters in the alphabet we're taken and now when you actually say what do we mean by the resulting change in F we should be writing as well where does it go after you nudge so when you take you know from that input point plus that nudge plus that little H what's the difference between that and the original function just or the original value of the function at that point so this top part is really what's represented d/f you know this is this is what's representing DF over here but you don't do this for any actual value of H you don't do it for any specific nudge the whole point largely the whole point of calculus is that you're considering the limit as H goes to zero of this and this is what this is what makes concrete the idea of you know tiny little notch or a tiny little resulting change it's not that it's any specific one you're taking the limit and you know you could get into the formal definition of a limit but it gives you room for rigor as soon as you start writing something like this now over in the multi variable world very similar story we can pretty much do the same thing and we're going to look at our original fraction and just start to formalize what we what we think of each of these variables is representing that that partial X still still it's common to use the letter H just to represent a tiny nudge in the X direction and now if we think about what is that what is that nudge and here let me draw it out actually the way that I kind of like to draw this out is you think of your entire input space as you know the XY plane if it was more variables this would be a high dimensional space and you're thinking of some point you know maybe you're thinking about as a B maybe I should specify that actually we're you know we're doing this at a specific point how you define it we're doing this at a very specific point a B and when you're thinking of your tiny little change in X you'd be thinking you know a tiny little nudge in the X Direction tiny little shift there and the entire function Maps that input space whatever it is to the real number line this is your this is your output space and you're saying hey how does that tiny nudge influence the output I've drawn this diagram a lot I'm just this loose sketch I think it's actually pretty good model because once we start thinking of higher dimensional outputs or things like that it's pretty flexible and you're thinking of this as your partial f partial x sorry your change in the X direction and this is that resulting change for the function but we go back up here and we say well what does that mean right if H represents that that tiny change to your x value well then you have to evaluate the function at at the point a but plus plus that H and you're adding it to the x value that first component just because this is the partial derivative with respect to X and the point B the point B just remains unchanged right so this is this is you evaluating it kind of at the new point and you have to say what's the difference between that and the old evaluation where it's just at a and B and that's it that's the formal definition of your partial derivative except oh I mean the most important part right the most important part given that this is calculus is that we're not doing this for any specific value of H but we're actually actually let me just move this guy give a little bit of room here yes but we're actually taking the limit here limit as H goes to 0 and what this means is you're not considering any specific size of DX any specific size of this really this is H you know considering the notation up here but any size for that partial X you're imagining that nudged shrinking more and more and more and the resulting change shrinks more and more and more and you're wondering what the ratio between them approaches so that would be the partial derivative with respect to X and just for practice let's actually write out what the partial derivative with respect to Y would be so I'll get rid of some of this one dimensional analogy stuff here don't need that anymore and let's just think about what the partial derivative with respect to a different variable would be so if we were doing it as partial derivative of F with respect to Y now we're nudging slightly in the other direction right we're noting in the Y direction and you'd be thinking okay so we're still going to divide something by that nudge and again I'm just using the same variable maybe you would be clearer to write something like the change in Y or to go up here and write something like you know the change in X and people will do that but it's less common I think people just kind of want the standard go-to limiting variable but this time when you're considering what is what is the resulting change oh and again I always I always forget to write in we're evaluating this at an at a specific point at a specific point a B and as a result maybe I'll give myself a little bit more room here so we're taking this whole thing divided by H but what is the resulting change in F this time you say F the new value is still going to be at a but the change happens for that second variable it's going to be that be b plus h so you're adding that nudge to the y-value and as before you you subtract off you see the difference between that and how you evaluate it at the original point and again the whole reason I move this over and gave myself some room is because we're taking the limit as this H goes to zero and the way that you think about this is very similar it's just that when you change the input by adding H to the y-value you're shifting it upwards so again this is this is the partial derivative the formal definition of the partial derivative looks very similar to the formal definition of the derivative but I just I just always think about this as spelling out what we mean by partial Y and partial F and kind of spelling out why it is that the Leibniz came up with this notation in the first place well I don't know if Leibniz came up with the partials but the the DF DX portion and this is good to keep in the back of your mind especially as we introduce new notions new types of multivariable derivatives like the directional derivative I think it helps clarify what's really going on in certain contexts great see you next video