The derivative is a way to show how steep a function is at a given point. Derivatives are similar to the slope of a line, but can be used for other curves as well. It is sometimes called the "instantaneous rate of change" of a function.
More specifically, the derivative is how much a function is changing at one given point. For functions that act on the real numbers, it is the slope of the tangent line at a point on a graph. The derivative is often written as \(\frac{dy}{dx}\) ("dy over dx" or "dy upon dx", meaning the difference in y divided by the difference in x). The d is not a variable, and therefore cannot be cancelled out. Another common notation is \(f'(x)\)—the derivative of function \(f\) at point \(x\), usually read as "\(f\) prime of \(x\)".
The derivative of y with respect to x is defined as the change in y over the change in x, as the distance between \(x_{0}\) and \(x_{1}\) becomes infinitely small (infinitesimal). In mathematical terms,
\(f'(a)=\lim\limits_{h\to 0}{\frac {f(a+h)-f(a)}{h}}\)
That is, as the distance between the two x points (h) becomes closer to zero, the slope of the line between them comes closer to resembling a tangent line.
A function's derivative can be used to search for the maxima and minima of the function, by searching for places where its slope is zero.
Derivatives are used in Newton's method, which helps one find the zeros (roots) of a function. One can also use derivatives to determine the concavity of a function, and whether the function is increasing or decreasing.
Calculus | Derivatives of a Function - Lesson 7 | Don't Memorise from Youtube Channel: Infinity Learn NEET