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Calculus for Machine Learning and Data Science
26h57m
So now that you know the scalar multiplication rule, the sum rule is just as simple. Imagine that you have a function f, and it expresses the sum of two functions g and h. So what is the derivative of f? Well, it's simply the derivative of g plus the derivative of h. I'd like to imagine the sum rule in the following way. Imagine that there's a boat, and there's a kid inside the boat. And the boat moves, and the child also runs inside the boat. So the boat moves a distance, let's say, of 10, and the child moves a distance of 2. And we're going to keep track with variables xb for the distance that the boat moves, and xc for the distance that the child moves. So if they're moving in the same direction, what is the distance that the child moves with respect to the Earth? Well, that distance is 12, because it's 10 plus 2. It's the distance that the boat moves plus the distance that the child moves inside the boat. Now that's with distances, but what happens with velocities? So let's do a quiz. Let's say that the speed of the boat is 0.6 meters per second, and the speed of the child inside the boat is 0.05 meters per second. And they're both traveling in the same direction. So here is a question. What is the speed of the child with respect to the Earth? So if you answered 0.65 meters per second, then you are correct, because velocities can be added if they are taken in the same direction. And that's pretty much the sum rule. It says if distances can be added, then velocities can also be added. It means that if functions can be added, then their derivatives can be added. So let's look at it with a plot. Here is a plot of time in the horizontal axis and distance in the vertical axis. And this is how the child moves inside the boat. This is how the boat moves. And this is the sum of the two, which is how the child moves with respect to the Earth. So let's look at some slopes. Here is some point in time, and we're going to take some small interval delta t, and we're going to calculate the average velocity over that interval for the child, for the boat, and for the sun. So for this, we take the horizontal step to be delta t, and the vertical step to be the distance. So these distances are going to be xc for child, xv for boat, and x total for the sum. Now we know that x total is xv plus xc, because these two distances add to x total. So if we divide everything by the same delta t, we obtain that the velocities add. So v total is equal to vb for the boat plus vc for the child. And now, all you need to do is let delta t go to zero, and you get the derivatives. So if you have three functions, f1, f2, and f1 plus f2, the derivative of f1 at a point is f1 prime of x. The derivative of f2 at that same point x is f2 prime of x. And what's the derivative of the sum? Well, since f is f1 plus f2, then f prime is f1 prime plus f2 prime. And the slope over here is going to be the sum of the slopes. And that's the sum rule.