## Introduction to the Definition of Derivative Calculator:

Definition of derivative calculator is an online tool that helps you to **evaluate the derivative** of a given function using the limit definition of derivative. It is used to find the rate of change of a function with respect to its variable of a given point.

The limit definition of the derivative calculator is a wonderful tool that helps you find the derivative of a complex function with multiple variables without using direct differentiation rules. This method is used in various fields like physics, engineering, and economics to find the rate of change.

## What is the Limit Definition of a Derivative?

The limit definition of a derivative is a process of finding the **instantaneous rate of change** of a function at a particular point on a graph in calculus. It gives information about the behavior of a function and infinitesimal change at that specific point.

It is mathematically defined as a function f(x) around a point a. The limit definition of derivative calculator uses the following formula,

$$ f’(x) \;=\; \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Whereas,

- f(x) is the function with respect to x variable
- f(x+h) is the rate of change of the function at point a
- h is the small infintesimal change

## How to Find the Derivative of a Function using the Definition?

To **find the derivative** of a function using the limit definition, the derivative using limit definition calculator uses the formula of the definition derivative because you cannot solve it with traditional differential rules. It has a specific method.

to find the derivative of a function f(x) using the limit definition, follow these steps that are given as:

**Step 1**:

Identify the given function f(x) and find the rate of change in function as f(x+h).

**Step 2**:

Add the f(x) and f(x+h) values in the formula of the definition of the derivative.

**Step 3**:

After putting the values, simply the given expression using an algebraic method.

**Step 4**:

If h is still present in the above expression then apply the limit so that h can be removed from the function.

**Step 5**:

After applying the limit again simplify it if it needed otherwise this is your solution for the definition derivative function.

## Solved Example of the Derivative of a Function:

An **example** of the derivative of a function is given below to let you understand the whole working process of the limit definition of derivative calculator.

### Example: find the derivative of:

$$ f(x) \;=\; \sqrt{x} $$

**Solution**:

The given function is,

$$ f(x) \;=\; \sqrt{x} $$

According to the definition, the rate of change of a function f(x) is,

$$ f(x + h) \;=\; \sqrt{x + h} \;and\; f(x) \;=\; \sqrt{x} $$

Put these values in the formula of derivative definition,

$$ f’(x) \;=\; \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

$$ f’(x) \;=\; \lim_{h \to 0} \frac{\sqrt{x + h} - \sqrt{x}}{h} $$

Use the rationalization method to simplify the above expression,

$$ =\; \lim_{h \to 0} \frac{\sqrt{x + h} - \sqrt{x}}{h} . \frac{\sqrt{x + h} + \sqrt{x}}{\sqrt{x + h} + \sqrt{x}} $$

$$ =\; \lim_{h \to 0} \frac{h}{h(\sqrt{x + h} + \sqrt{x})} $$

$$ =\; \lim_{h \to 0} \frac{1}{(\sqrt{x + h} + \sqrt{x})} $$

For the calculation of h apply the limit and the solution you get is,

$$ =\; \frac{1}{2 \sqrt{x}} $$

## How to Use the Definition of Derivative Calculator?

The limit definition of the derivative calculator has an easy-to-use interface, so you can easily use it to evaluate the derivative of a given limit definition derivative function. Before adding the input for the solutions, you must follow some simple steps. These steps are:

- Enter the given derivative of a limit function in the input box.
- Choose the variable of the derivation of the limit function in its respective field.
- Recheck your input value for the derivation of the definition function before hitting the calculate button to start the calculation process in the derivative using limit definition calculator.
- The “
**Calculate**” button gives the desired result of your given limit derivative definition problem. - If you want to try out our derivative definition calculator to check the accuracy of the solution then use the load example to get a clear understanding of its works.
- The “Recalculate” button brings you to a new page for solving more limit definition derivation questions.

## Output of Limit Definition of the Derivative Calculator:

The limit definition of derivative calculator gives you the **solution** to a given differential problem when you give the input and it provides you with solutions. It may contain as:

**Result Option**:

When you click on the result option as then the definition of a derivative calculator provides you with a solution of limit definition of derivation questions.

**Possible Step**:

When you click on the possible steps option then the derivative limit calculator gives you a solution with steps.

## Benefits of Using the Limit Definition Calculator:

The derivative as a limit calculator gives you multiple **benefits** whenever you use it to calculate the limit derivative of the given function and you get its solution. These benefits are:

- The definition of a derivative calculator saves the time and effort that you consume in solving complex derivative definition questions.
- It is a free-of-cost tool that provides you with a solution for a given derivative of a limit definition problem to find results without spending.
- The derivative definition calculator is an adaptive tool that allows you to calculate the different types of limit definitions of derivative functions (logarithmic, exponential, algebraic, inverse trigonometry, etc).
- You can use this calculator for practice to get a strong hold on the concept of the limit definition of derivative.
- The limit definition of the derivative calculator gives you an accurate solution of limit the definition of differential problems without any difficulty.
- The limit definition of derivative calculator with steps can operate on an online platform which means you can use it anytime without any limit.