 In this video, I want to talk a little bit about how derivatives are important to the study of biology. In biology, we talk about things like blood flow, the law of laminar flow, and velocity gradients. There's a lot of jargon I can throw out there, but one example I think that's very approachable for students in a calculus one setting would be the idea of population growth. So imagine we have N to be the population of some culture of organisms, plant, animal, fungi, whatever. Whereas the population with respect to time, so N equals F of t is a population with respect to time. So over time the population changes, maybe goes up or goes down based upon various factors. So what would something like delta N represent here? Well, if we take delta N to be the change of N from time stamp t1 to time stamp t2, we're taking the difference of F of t2 with F of t1. F of t1 would be the population at time t1. F of t2 would be the population at the time stamp t2. And so delta N that is measuring the change of population from time t1 to time t2. So this implies that the average rate of growth during the time interval t1 to t2 would look like delta N over delta t. We would take delta N, which is the change of population from t1 to t2, divide that by the length of time. Like is this a year? Is this a month? Is this a decade? The average rate of growth would then measure how the population grew on average over that time interval. But we might be interested not just in the average rate, but we can be interested in the instantaneous growth rate of that population. It's like, great, okay, I see how that thing is going to be growing over one year, two years, 10 years, 100 years. But how rapidly is the growth rate changing right now? Like how fast is it growing at this moment? If we allow the change of time to go to zero, the growth rate would then be the limit as delta t goes to zero of delta N over delta t, which gives us then dN over dt. This gives us the growth rate of the population at a specific instance of time. To be more specific, consider populations, let's say bacteria, right? Which bacteria often grow so rapidly that we need to know at this instance what's going on. So let's imagine that a bacteria culture has a homogeneous nutrient medium. That is to say every organism has equal access to vital nutrients. So all portions of the petri dish have equal opportunity to grow. There's no like inequity inside of our culture of bacteria whatsoever. Now, suppose by sampling, the population at certain intervals is determined that the population doubles every hour. So this bacteria doubles its population every hour. Very fast growing bacteria here. If the initial population is given as some N sub zero, so do we start off with five bacteria, seven bacteria, a million bacteria, a single bacterium. What's going on here? If we have this initial population, that tells us that then f of t will look like N sub zero times two to the t, right? The idea here is that if it doubles every hour and tees the number of hours, well, one hour, it means it's doubled. Two hours means it's doubled twice, so you times it by four. If it's been three hours, it's doubled three times, which times it by eight. So that's just this basic exponential growth one could expect. So if the growth rate of the bacteria population at time t, what would that be? Well, it's going to be D and over dt. We take the derivative of this function right here, which we know how to calculate the derivative of an exponential function. This constant multiple of N of t can come out because the initial population will not change over time. The initial population is what the population was at t equals zero. That doesn't change. It's constant, so we can factor it out. We take the derivative of two to the t because this is an exponential whose base is not e. We would get back the original exponential two to the t, but we also have to pay the tariff of the natural log of two. So the derivative is just you have this extra factor of the natural log of two. And so, for example, suppose we start with an initial bacteria population of 100, then what would be the growth rate after four hours? We want to compute the derivative at t equals four. Well, based upon the formula N zero was equal to 100. So we plug that in there t equals four. So we plug that in there to the fourth will be 16 16 times 100 to 1600. So we take 1600 times the natural log of two. That's approximately 1109. That is 11109. This means that after four hours, the bacteria is growing at a rate of 1109 back 1100 essentially was rounded. It's going to rate 1100 bacteria per hour. So that means at four hours, it's like if you're adding 1100 new bacteria every hour. But of course, the more and more you get, the faster this is going that this function, which exponential growth has also exponential growth rate. And so as the more and more this bacteria has to grow, the faster it's going because the derivatives measuring at this hour at what rate are we increasing? And at four hours, it's increasing at a rate of about 1100 bacteria per hour.