The two commonly used ways of writing the derivative are Newton's notation and Liebniz's notation. Newton's notation involves a prime after the function to be derived, while Liebniz's notation utilizes a d over dx in front of the function. These two methods of derivative notation are the most widely used methods to signify the derivative function.
I want to talk about derivative notation, there are two main forms of derivative notation, there's the Newton form and the Leibniz form. These two forms are named for the two co creators of Calculus. Now the Newton form is the one we've been using so far it's the so called prime form. Now let's suppose we have a function f of x, the Newton form of derivative is f prime of x. If we have a function y like y equals x squared we would say its derivative is y prime. And if we wanted to talk about just the expression, x squared plus 1 you could write x squared plus 1 in parenthesis prime and all of these mean the derivative. But Leibniz form works a little differently, there is this notation here d over dx this is called the differential operator and what it basically means is the derivative with respect to x of f of x or the derivative with respect to x of y or the derivative with respect to x of x squared plus 1. What's great about this is you see the operation of differentiation with Leibniz form and it has a little shorter version when you have the derivative with respect to x of y you can write that dy dx and this is something you'll see a lot of or df dx you'll occasionally see. Now one of the things that this highlights is that differentiation or the derivative is an operation that you perform on a function. And I want to highlight the difference between the derivative of a function and the process of differentiation. Differentiation is the process of getting the derivative. So let's imagine that here's a function and this is the differentiation machine the process of getting the derivative. The result is the actual derivative, so remember that differentiation is the process you differentiate a function and the result is the derivative f prime of x.