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:6:55

hello everyone so I have here the graph of a two variable function and I'd like to talk about how you can interpret the partial derivative of that function so specifically the function that you're looking at is f of XY is equal to x squared times y plus sine of Y and the question is if I take the partial derivative of this function so maybe I'm looking at the partial derivative of F with respect to X and let's say I want to do this at negative 1 1 so we'll be looking at the partial derivative at a specific point how do you interpret that on this whole graph so first let's consider where the point negative 1 1 is if we're looking above this is our x axis this is our Y axis the point negative 1 1 is sitting right there so negative 1 move up 1 and it's the point that's sitting on the graph and the first thing you might do is you say well when we're taking the partial derivative with respect to X we're going to pretend that Y is a constant so let's actually just go ahead and evaluate that so when you're doing this it looks it says x squared looks like a variable Y looks like a constant sine of Y also looks like a constant so this is going to be we differentiate x squared that's 2 times x times y which is like a constant and then the derivative of a constant there is 0 and we're evaluating this whole thing at X is equal to negative 1 and Y is equal to 1 so when we actually plug that in it'll be 2 times negative 1 multiplied by 1 which is 2 negative 2 excuse me but what does that mean right we evaluate this and maybe you're thinking this is kind of a slight nudge in the X Direction this is the resulting nudge of F what does that mean for the graph well first of all treating Y as a constant is basically like slicing the whole graph with a plane that represents a constant Y value so you know this is the y-axis and the plane that cuts it perpendicularly that represents a constant Y value this one represents the constant Y value 1 but you could imagine it you know you could imagine sliding the plane back and forth and that would represent various different Y values so for the general partial derivative it you know you can imagine what one you want but this one is y equals one and I'll go ahead and slice the actual graph at that point and draw draw a red line and this red line is basically all the points on the graph where Y is equal to one so I'll just kind of emphasize that where Y is equal to one this is y is equal to one so when we're looking at that we can actually interpret the partial derivative as a slope because we're looking at the point here we're asking how the function changes as we move in the X direction and from single variable calculus you might be familiar with thinking of that as the slope of a line and to be a little more concrete about this I could say you know you're starting here you consider some nudge over there just some tiny step I'm drawing it you know as a sizable one but you imagine that is a really small step as your DX and then the distance to your function here the change in the value of your function as your essid DX but I should say partial X or del X partial F and as that tiny node gets smaller and smaller this this change here is going to correspond with what the tangent line does and that's why you have this this rise over run feeling for the slope and you look at that value and the line itself looks like it has a slope of about negative two so it should actually make sense that we get negative two over here given what we're looking at but let's do this let's do this with a partial derivative with respect to Y let's erase what we've got going on here and I'll go ahead and move the graph back to what it was get rid of these guys so now we're no longer slicing with respect to Y but instead you know let's say we slice it with a constant X value so this here is the x-axis this plane represents the constant value x equals negative one and we could slice the graph there you could kind of slice it I'll draw the red line again that represents the curve and this time that curve represents the value X or equals negative one it's all the points on the graph where x equals negative one and now when we take the partial derivative we're going to interpret it as a slice as the slope of the as a slope of this resulting curve so that slope ends up looking like this that's our blue line and let's go I hadn't evaluate the partial derivative of F with respect to Y so I'll go over here use a use a different color so the partial derivative of F with respect to Y partial Y so we go up here and it says okay I see x squared times y it's considering x squared to be a constant now so it looks at then says X Y or a constant while your the variable constant times the variable the derivative is just equal to that constant so that x squared and over here sine of Y the derivative of that with respect to Y is cosine Y cosine Y and if we actually want to evaluate this at our point negative 1 1 what we'd get is negative 1 squared plus cosine of 1 and I'm not sure what the cosine of 1 is but it's something a little bit positive and the ultimate result that we see here is going to be you know 1 plus something I don't know what it is but it's something positive and that should make sense because we look at the slope here and it's a little bit more than one not sure exactly but it's a little bit more than 1 so you'll often hear about people talking about the partial derivative as being the slope of the slice of a graph which is great if you're looking at a function that has a two variable input and a one variable output so that we can think about its graph and in other contexts that might not be the case maybe it's something that has a multi-dimensional output we'll talk about that later when you have a vector-valued function what it's partial derivative looks like but maybe it's also something that has you know 100 inputs and you certainly can't visualize the graph but the general idea of saying well if you take a tiny step in the direction here I'll actually walk through it in this graph context again you know you're looking at your point here and you say we're going to take a tiny step in the Y direction and I'll call that partial Y and you say that makes some kind of change it causes a change in the function which you'll call partial F and as you imagine this getting really really small and the resulting change also getting really small the rise over run of that is going to give you the slope of the tangent line so this is just one way of interpreting that ratio the change in the output that corresponds to a little nudge in the input but later on we'll talk about different ways that you can do that so I think graphs are very useful when I move that the the text doesn't move I think graphs are very useful for thing about these things but they're not the only way and I don't want you to get too attached to graphs even though they can be handy in the context of two variable input one variable output see you next video