How to Find the Derivative of an Integral? [Simple Steps]

Calculus is one of the most important parts of mathematics that we come across in high school or exactly in classes 11th and 12th. Calculus is the composition of two main components: Integration and Differentiation. Differentiation refers to the process of finding the rate of change of an object. Integration is the exact opposite of that. It is also known as anti-differentiation. In layman’s term, Integration is the process of finding things or bringing them close together. It is used to describe displacement, area, volume, etc by combination of infinite data. Today, we will tell you how to find the derivative of an Integral.

Applications of Integrals

Integration is an important concept used widely in the scientific world. Engineers use Integrals to find out the total amount of materials necessary for constructions. E.g. constructing a dome over a park. It is an extremely important concept for architects in several situations. Integration is useful not just to improve the architecture of buildings, but also of important infrastructures like bridges.

Integration helps engineers determine the exact length of cable required to connect two substations far away from each other.

In the world of physics, scientists use Integration to determine the Centre of Mass, Centre of Gravity, Mass Moment of Inertia of a sports utility vehicle. It also helps to calculate velocity and trajectory of an object, predict the position of planets, and understand electromagnetism.

Integration also helps in the operations of a research analyst. A research analyst uses calculus when observing different processes at a manufacturing corporation. By considering the value of different variables, they can help a company improve operating efficiency, increase production, and raise profits.

Statisticians use Integration to help them develop an idea to evaluate survey data for developing business plans for companies. A survey contains questions that can receive different answers from different audiences. Integration, as well as Differentiation, helps a scientist in determining an accurate prediction.

See also: Top Important Derivative Rules to Simplify Your Calculus

How to Find the Derivative of an Integral

Finding Derivative of an Integral is an easy topic and one of the scoring topics in the exam. Read to the end and you will definitely learn how to find the derivative of an Integral. We know that a derivative of a function ‘y’ is:

Y = dy/dx

Now, we will give you a systematic explanation of how to find the derivative of an Integral. We will understand the process with the help of an example. So, let us take:

  • The first thing to understand here is that f(x) is an accumulative function. This means that it is finding the area under a curve from a number x to t.
  • Since it is an accumulation function and we want to find it’s derivative, we are taking derivative of an integral from a number to x, we will the first fundamental theorem of calculus.
  • The first fundamental theory of Calculus states that when we take the derivative in terms of X, of an integral from a constant up to X, we just get back the original function on the inside. However, the variable changes from T to X.

Thus, derivative of the function (x) will be:

Understand that we will not change anything about the inside of the function, except change the T to X. Well, here it is. This is how to find the derivative of an Integral.


If you found it difficult to understand, let us take another example to let your mind get around the idea of how to find the derivative of an Integral.

Here, we almost have an accumulation function and we can almost solve it using the first fundamental theorem of calculus.

To solve this, we will apply the rule for definite integrals that allows us to flip the boundaries, interchanging A&B. However, to do that we have to put a negative sign out in the front. Thus, after flipping we get the integral:

Finally, let us give a try to difficult and quite a tricky question. If you can solve this one you will be able to find out the derivative of an integral. This question will test your complete knowledge of the topic differentiation.

Try it. If you find yourself unable to solve it, here is how to do it:

This function is also quite not an accumulation function because of the x cubed. So, we think of it as a composition. Thus, consider:

Now, we will solve the equation with the help of the chain rule.

The chain rule says that we take the derivative of f, we plug in x cubed, and then we multiply that by the derivative of x cubed, which would be 3x squared.


Well, if you understood these few examples properly, you will be able to solve any integral without any difficulties. In this post, we have told you how to find the derivative of an Integral. Children find the derivative of an integral to be an extremely difficult topic. However, with the correct use of theorems and rules, this is one of the easiest concepts you can understand. Well, we hope you liked the post. If you have any other suggestion for us, drop it in the comment section.

See also: What is Implicit Differentiation? Easily Explained with Examples

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