One of the building blocks of calculus is finding derivatives. In fact, we have a separate name for it and it is called as differential calculus. Differential calculus is the branch of calculus that deals with finding the rate of change of the function at a given point. The other significant subdivision of calculus is integral calculus which is related to the differential calculus in a way that the process of finding the derivatives and integrals are reverse of each other. The process of finding derivatives is known as differentiation and the process of finding integrals is known as integration. But in this article, we will only discuss the derivatives and how to interpret them geometrically.
Before proceeding, you must know that to understand the geometrical representation of derivatives, you should be familiar with how to differentiate the function, differential rules and finding derivatives using the limit formula. The reason behind it is that the geometric representation of the derivatives is a higher level concept and to ace it, your fundamental concepts related to differentiation must be cleared.
Derivative means the rate of change in one variable with respect to another variable. In other words, we can say that a derivative of the function is an instantaneous rate of change of the function at a given point. The derivatives are mathematically represented as follows:
Alternatively, we also use the following notation to represent derivatives of a function:
We can take as many derivatives of differentiable functions as possible. When we take the derivative of the original function, it is known as the first derivative. Similarly, when we further differentiate the first derivative, we get a second derivative and so on. We call second, third and fourth derivatives of the function higher order derivatives. We know that differentiation takes the original function as an input and then returns a derivative as an output.
We know that the graphs are used in mathematics to interpret the patterns or relationships between two variables. The graph of a function changes after differentiating a function. When we differentiate the first derivative further to find the second derivative, the graph changes again.