Interpreting the meaning of the derivative in context

we're told that Eddie drove from New York City to Philadelphia the function D gives a total distance Eddie has driven in kilometers T hours after he left what is the best interpretation for the following statement D prime of 2 is equal to 100 so pause this video and I encourage you to write it out what do you think this means and be sure to include the appropriate units all right now let's do this together if D is equal to the distance the distance driven then to get D prime you're taking the derivative with respect to time so one way to think about it is it is the rate of change of D so we could view this as D prime is going to give you the instantaneous instantaneous instantaneous rate and they are both functions of T so one way to view D prime of 2 is equal to 100 that would mean well what is our time now well that is our T and that's in hours so 2 hours actually let me color code it so 2 hours after leaving after leaving Eddie Eddie drove drove and this means let me be grammatically correct drove at an instantaneous instantaneous instantaneous rate of and let me use a different color now for this part of 100 and what are the units well the distance was given in kilometers and now we're going to be thinking about kilometers per unit time kilometers per hour so this is 100 kilometers kilometers per hour so that's the interpretation there let's do another example here we are told a tank is being drained of water the function V gives the volume of liquid in the tank in litres after T minutes what is the best interpretation for the following statement the slope of the line tangent to the graph of V at T equals seven is equal to negative three so pause this video again and try to do what we just did with the previous example write out that interpretation and make sure to get the unit's right all right so let's just remind ourselves what's going on V is going to give us the volume as a function of time volume is in liters and time is in minutes and so if they're talking about the slope of the tangent line to the graph the slope of the tangent line to the graph of V that's just V Prime so if you take the derivative with respect to time that's going to give you V Prime and these are all functions of T these are all functions of T and they say at T equals seven it's equal to negative three so this which is the same thing as the slope of tangent line slope of tangent tangent line and they tell us that V prime of at time equals seven minutes our rate of change of volume with respect to time is equal to negative three and so you could say if we were to write it out this means that after after seven minutes seven minutes the tank is being drained at an instantaneous instantaneous that's why we need that calculus for that instantaneous rate and an instantaneous rate of now you might be tempted to say it's being drained an instantaneous rate of negative three liters per minute but remember the negative three just shows that the volume is decreasing so one way to think about it is this negative is already being accounted for when you're saying it's being drained if this was positive that means it is being filled so it is being drained at an instantaneous rate of three liters per minute three liters per minute and how did I know the units were liters per minute well the volume function is in terms of leaders and the time is in terms of minutes and then I'm taking the derivative with respect to time so now it's going to be liters per minute and we are done