Derivatives are Confusing — So Here’s a 3 Minute Explanation

Hooray for calculus!

You have probably heard about the central principle of mathematical functions if you have studied linear regression. You can articulate this with the following example — assume that you have used the number of bathrooms in a house as a predictor and the house rental price as the target variable. The mathematical function of this example would be rental price = f(bathrooms), or more generically, y = f(x). Then, let’s assume that the price of the apartment is set in a very simplistic manner and the relation between the number of bathrooms and the rental price is completely linear. Assume that, for every bathroom that is provided, the price rises by \$400 per month. We can express the price in that situation as follows: price = 400 ∗ number of bathrooms, or, 𝑦 = 𝑓(𝑥) = 400 ∗ 𝑥 = 400𝑥. Remember also that the intercept is not here. Thus, as the number of bathrooms shifts — either higher or lower — we want to look further into how the price of rent changes. In this blog post, I am going to explain what a derivative is, how to determine a linear function’s derivative, and describe how derivatives are the instantaneous rate of a function’s change. I will concentrate on computing the derivatives of straight line functions (aka a linear function) only.

Let’s add a new example in order to introduce derivatives. Suggest that we want a function that reflects an individual who is walking. This example allows one to see how distance varies in response to time, or in other words, the walker’s speed. Derivatives are significant because they inform us how at any particular moment a function is evolving. Here, when we inquire about the rate of change — which is the main principle behind derivatives — we essentially ask how quickly the walker is moving.

We will see where a person is at a given time, then wait an hour to see how far they have gone to measure their speed in miles per hour. Technically, we can also wait two hours, then split the time travelled by two. Because this example has a perfectly linear function, our strategy is to divide the amount of miles travelled by the number of hours spent walking. By seeing where they begin at hour zero and finish at hour one, we calculate the speed of our walker. Miles per hour, thus, is miles/hour = (end distance-start distance) / (end time-start time) = (3–0) / (1–0) = 3. As our jogger is traveling at a steady pace, the derivative is also our measured rate of change of 3 miles per hour.

Just one indication of the rate of change is miles per hour. Whenever we come across the term per, we realize that this is a sort of rate of change. Every rate of change is determined in the same way — the change in the y-axis value is divided by the change in the x-axis value. Changes in y or x are often described with the Greek symbol delta (Δy, Δx). If we are given a f(x) function, we say that f ’(x) — interpreted as “f prime of x” — is the derivative of that function.

*Note* In general, we can assume that a straight line’s derivative is equivalent to the “rise over the run” — aka the slope.

And that’s it — the core concepts behind derivatives. Derivatives do get much more complicated and much more dense, but if you’re able to engrain in your brain the things presented in this blog post, then handling that complexity will become much, much easier. Hopefully this helped you with just that.