I don’t mean in which cases I can use them, nor how they work. I know how they work (at least at a basic level, the derivative of ax\^b is abx\^(b-1), but I mean… why is a function that does those steps useful to solve any problem? It really seems like a random choice of operations.
In: Mathematics
The example you gave is specifically for polynomials. Those steps are only useful because, with specifically polynomials, this represents the rate of change in that polynomial.
There is no ax^b for a function like sin(x) yet its derivative is way *more* useful.
There is a broader test to use for derivatives, and this involves taking the change over time (rise over run) as the run gets smaller. What does the rise over run get closer and closer to as the run gets closer to zero? This definition tells us, more or less, the slope of the line at a point. How one number changes in response to another number. That’s important anywhere that change is happening.
It’s actually extremely useful in specific areas. If you have an expression like the above, it can represent the amount or state of something. If you take its derivative, you get an expression for the rate (and direction) of the *change* of that state or amount. You can actually make some important inferences on systems (represented by functions/expressions) or variables simply by looking at its derivative.
It only seems random because you’re being given functions that are made out of thin air, so what they represent is meaningless, and the process of knowing how to take a derivative is their only point to exist.
My mums house is 60 miles away and it takes me an hour, so on average I do 60mph on the journey there.
BUT if you have an equation that tells you where a car is along the road, you can get the derivative of that equation which will tell you it’s speed
So it’s slightly more detailed information than ’60mph’ which is what keeps my mum happy. She doesnt have the equation and doesnt know I did 80mph for some bits
The derivative itself is useful.
Because you get to easily see the rate of change of something. And for higher derivatives you get to see the rate of change of the rate of change.
The derivative of a function is its instantaneous rate of change. That’s often a very important property of a system.
If you have a moving object, the first two derivatives of its position are its velocity and acceleration. If you know some other property that relates to its velocity or acceleration, like the force on it, you’re on your way to something useful.
Consider a mass on a spring. As it moves back and forth, the force exerted by the spring depends on the compression of the spring, which depends on the position of the mass. So now you can relate the force (acceleration) to the position, which gives you a simple differential equation. You can solve the differential equation to describe how the mass moves, and you can even incorporate external forces and damping.
There are a whole bunch of situations where you know about how something changes, and you want to model that something over time. Temperature is another good one. If you have a heat source, you can figure out how fast heat is moving through the system, which is a rate of change. You want to know the temperature itself, so now you’re working with derivatives.
In physics you use derivatives all the time. Speed is derivative of position over time, acceleration is derivative of speed over time.
With derivative you can have a simple test of smaller or greater than 0 to see if derived function is increasing or decreasing. Derivative can be used to find local minimum and maximum of a function which has use in optimisation problems.
I would add to other comments that the specific derivative you mention (ax^b -> abx^(b-1)) is especially useful as if you look at any smooth curve and zoom in, it will start to look like the curve of some function ax^0 + bx^1 + cx^2 ….
Here’s an image to demonstrate [image](https://cocalc.com/blobs/projects/a8975d68-235e-4f21-8635-2051d699f504/.sage/temp/compute4-us/22699/tmp_zosLMe.svg?uuid=11530e5d-dd02-4f67-bdd3-4692f24b82bb)
The red line in the plot is a simple function of the form above which approximates the more complex blue one in the region of the marked point. The dotted line is an even simpler approximation of the same form.
No matter how complex the original function, we can use the fact it looks like this simpler function to find it’s derivative using the formula you gave. This allows us to do calculus with ‘real world’ functions that we can’t necessarily write a nice equation for and differentiate algebraicly.
A function shows the value of something at certain parameters.
A derivative of a function shows the rate of change for that function in respect to a certain parameter (Time, for example). An integral shows the sum of the function in respect to a certain parameter (again, like Time).
A simpler example is if you have a function that describes the position of something, then the derivative of this function would be the rate of change in **position**. Rate of change of position? That’s **speed**. Suddenly, you have a function that describes the speed.
If you derive this function of speed, now you get the rate of change in speed. That just means **acceleration**.
Derivative (and Integral) is incredibly useful and is sometimes the basis of certain smaller fields, from robotic control systems, to stock market exchanges.
Derivative function measure the rate of change of something. Which is useful in many areas.
An object acceleration is the change rate of its speed. That’s a derivative.
It also ties correlated items that change together.
If you have to pay a booking fee for something, the fee changes following the price changes of the underlying item you want to buy. That’s a derivative too. (Example taken from financial options).
If the derivative equals 0, there is no change to the vertical axis when the horizontal axis change.
If the derivative equals X, one increment on the horizontal axis will cause X increments on the vertical axis.
First, the derivative of x^n = n x^(n-1) thing is not what the derivative actually *is*. That’s a special case of how to calculate it in certain circumstances. The derivative is a method of calculating rates of change. The derivative of position (where you are) is velocity (how fast you’re going). The derivative of how much water is in your tank is how fast water is entering/leaving your tank.
The equation that defines the derivative is the limit definition: limit as change in x approaches 0 of f(x+change in x) – f(x) over change in x. That is what the derivative is: how much things change over a period of time that approaches 0. All the other rules that you will learn, including the power rule you mentioned, are just tricks to make it easier to compute.
But what is it useful for?
Let’s say you throw a ball straight up in the air. How high will it go? Well, one way to figure it out is to say that the highest point is when it stops going up and starts going down. What does that mean in terms of math? Well it means it’s velocity switches from positive (up) to negative (down) – which means it passes through zero. Velocity is the derivative of position. So the highest point of position is when the derivative of position is zero. This type of thing matters to ballistics and similar.
Cool. But you don’t throw balls for a living, you sell candy bars, so why do you care about throwing balls? Well, you might notice that the more you charge for your candy bars, the fewer people buy them. These candy bars cost you $1, and you’ve run a survey that says that you’ll sell 1000 per month if you sell for $2, and that for every cent you increase the price, you lose exactly one sale per month. What price will get you the most profit?
That’s the same problem as throwing the ball: you can easily create an equation for the profit that you’ll make as a function of price. The maximum profit happens when increasing the price switches from increasing profit to decreasing profit.
Or let’s say you have a computer, and you want it to learn to recognize pictures of cats. If something is a picture of a cat, it should output 1. If not, then 0. Well, what you can do is make a function with a whole bunch of random pieces in it, and shove a picture of a cat through it. It will give you a random number, say 0.5. You then use derivatives to figure out how to change the random pieces of your function to make that output a little bit closer to 1. Then repeat this a millions of times with different pictures of cats and non cats, and this is machine learning.
The applications are endless. I’ve only touched on what is called optimization problems, which is just one corner of what derivatives are used for.
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