Definition of Derivative Calculator

Do you want to determine the derivative of a function using the limit definition of derivative? Use the definition of derivative calculator and get an accurate results for free.

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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.

Definition of Derivative Calculator with Steps

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.

Related References

How would you define definition of derivative?
Explain the evaluation process of definition of derivative:
Give some examples of definition of derivative:
Give an Overview of definition of derivative?

Frequently Ask Questions

When to use the definition of derivative

The limit definition of derivative is used in various scenarios where derivative rules may not be applicable with the limit definition of derivative.

When dealing with complex functions to find the concept of the derivative as a rate of change, the velocity with respect to time t. While derivative rules are efficient for many functions, understanding the limit definition of derivative to ensures the understanding of calculus in certain analytical, educational, and non-standard function scenarios.

Can you find tangent without using the definition of derivative

Yes, you can find the equation of a tangent line to a curve at a given point without explicitly using the definition of derivative in its limit form.

While this method uses the derivative method to find f′(x) to determine the slope of the tangent line, but it does not use the limit definition of the derivative.

How to find derivative of a trigonometric function using the definition

To find the derivative of a trigonometric function f(x) using the limit definition of the derivative, let's consider the example of f(x) = sin⁡(x).

Solution:

The formula of definition derivative for a function,

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

Put f(x) = sinx ,f(x + h) = f(sin x + h) in the above formula,

$$ For f(x) \;=\; sin(x) $$

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

Apply the trigonometric formula to simplify the above expression,

$$ f’(x) \;=\; \lim_{h \to 0} \frac{2cos (\frac{x + h + x}{2}) sin (\frac{x + h - x}{2})}{h} $$

$$ f’(x) \;=\; \lim_{h \to 0} \frac{2 cos(x + \frac{h}{2}) sin(\frac{h}{2})}{h} $$

$$ f’(x) \;=\; \lim_{h \to 0} \frac{2cos(x + \frac{h}{2})sin (\frac{h}{2})}{h} $$

$$ f’(x) \;=\; \lim_{h \to 0} \frac{2cos(x)sin( \frac{h}{2})}{h} $$

Apply the limit to remove the h,

$$ f’(x) \;=\; \lim_{h \to 0} \frac{2cos(x) . \frac{h}{2}}{h} $$

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

$$ f’(x) \;=\; \lim_{h \to 0} cos(x) $$

The result of given function is,

$$ f’(x) \;=\; cos(x) $$

How to find derivative of 1/x using limit definition

To find the derivative of f(x)=1/x using the limit definition of the derivative, follow these steps:

The formula of limit definition derivative of f(x) at a point x = a is given by:

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

The function is given as f(a) = 1/a and f(a + h) is 1/a+h as x = a, put in above formula,

$$ For\; f(x) \;=\; \frac{1}{x}: $$

$$ f’(a) \;=\; \lim_{h \to 0} \frac{\frac{1}{a + h} - \frac{1}{a} }{h} $$

Simplify it using an algebraic method,

$$ f’(a) \;=\; \lim_{h \to 0} \frac{\frac{1}{a + h} - \frac{1}{a}}{h} \;=\; \lim_{h \to 0} \frac{\frac{a - (a + h)}{a (a + h)} }{h} $$

Use the Lcm method to solve it,

$$ f’(a) \;=\; \lim_{h \to 0} \frac{\frac{1}{a + h} - \frac{1}{a} }{h} $$

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

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

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

To eliminate the h, apply the limit

$$ f’(a) \;=\; \lim_{h \to 0} \frac{-1}{a(a + h)} $$

The result of a given function,

$$ f’(a) \;=\; \lim_{h \to 0} \frac{-1}{a(a + h)} \;=\; \frac{-1}{a^2} $$

How do I use the limit definition of derivative to find f'(x) for f(x)=√x+3 ?

To find f′(x) for f(x)=√x+3 using the limit definition of derivative, follow these steps.

The formula of the derivative of f(x) at a point x = a is given by:

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

The function is given as f(a) = √a+3 and f(a+h) is √a + 3 + h as x = a, put in the above formula. Simplify it using an algebraic method,

$$ f’(x) \;=\; \lim_{h \to 0} \frac{\sqrt{x + h + 3} - \sqrt{x + 3}}{h} $$

Rationalize the above function,

$$ f’(x) \;=\; \lim_{h \to 0} \frac{\sqrt{x + h + 3} - \sqrt{x + 3}}{h} . \frac{\sqrt{x + h + 3} + \sqrt{x + 3}}{\sqrt{x + h + 3} + \sqrt{x + 3}} $$

$$ f’(x) \;=\; \lim_{h \to 0} \frac{(x + h + 3) - (x + 3)}{h(\sqrt{x + h + 3} + \sqrt{x + 3})} $$

$$ f’(x) \;=\; \lim_{h \to 0} \frac{h}{h(\sqrt{x + h + 3} + \sqrt{x + 3})} $$

$$ f’(x) \;=\; \lim_{h \to 0} \frac{1}{\sqrt{x + h + 3} + \sqrt{x+3}} $$

Apply the limit,

$$ f’(x) \;=\; \frac{1}{\sqrt{x+3} + \sqrt{x+3}} $$

$$ f’(x) \;=\; \frac{1}{2\sqrt{x + 3}} $$