Derivative at a Point Calculator

Introduction to the Derivative at a point Calculator

The point differential calculator is an online freely available tool for finding the derivative of the given function at a particular point. It is a tool that compute the slope of the line tangent at some specific point.

derivative at a point calculator with steps

The derivative of a function at a point calculator makes the calculations easier and faster. It evaluates the derivatives in no time and gives you error-free results. This makes it easier to find high-order derivatives.

What do you know about the Derivative Calculator at a point?

A point differential calculator is an online tool that calculates the momentous change in the function or the change in slope of a tangent at a particular point.

In this calculator, the derivative of the given function at a specific point is calculated. The derivative at point calculator finds the average value of that given function at a particular point, allowing you to not only determine the slope or rate of change but also gain insights into the behavior of the function at its extreme points. Click here and try our extreme value calculator.

Formula for the Derivative at a Point Calculator

The derivative at a point calculator is a tool that uses the same formula used in the differentiation calculator. The following are the formulas used for derivative functions at different points.

$$ f'(x) \;=\; \lim\limits_{Δx \to 0} \frac{f(x+Δx)-f(x)}{Δx} $$

at some point x=a

$$ f'(a) \;=\; \lim\limits_{Δx \to a} \frac{f(x)-f(a)}{x-a} $$

This the the formula used in the derivative at point calculator for doing all of its calculations.

Let's get this concept with a following example:

Example: Find the derivative of a function f(x) = x³ at any point x.

Solution: According to the above-mentioned formula:

$$ f'(x) \;=\; \lim\limits_{t \to x} \frac{f(t)-f(x)}{t-x} $$ $$ f'(x) \;=\; \lim\limits_{t \to x} \frac{t^3-x^3}{t-x} $$ $$ f'(x) \;=\; \lim\limits_{t \to x} \frac{(t-x)(t^2+tx+x^2)}{t-x} $$ $$ f'(x) \;=\; \lim\limits_{t \to x} (t^2+tx+x^2) $$

putiing the limit t approaches to x, this will implies

$$ f'(x) \;=\; (t^2 + x \cdot x + x^2) $$ $$ f'(x) \;=\; 3x^2 $$

Hence, this will show that the derivative of a function x³ at some point "x" is 3x².

Related: Also try our calculator second derivative for differntiate the above function twice.

How does the Derivative of function at a point Calculator Works?

To find the Point Differential Calculator, usually following steps are performed to calculate the derivative of given functions and the slope of the tangent:

Step 1: Open the calculator page, and put the function in "Enter Function".

Step 2: Now select the order of derivative from the drop-down list i.e 1st,2nd,3rd,..., how many times you want to differentiate the function.

Step 3: Then put the value of a point "x" at which you wish to take the derivative (derivative at a point).

Step 4: After entering all the values just click on the "CALCULATE" button. The derivative calculator at a point will evaluate the slope of the tangent of the given function.

Finally, the results of your given differential results are displayed on your screen. After evaluating the differential function, refresh the page for new calculations.

Note: Explore our range of calcuoators beyond the derivative at point calculator i.e approximate value calculator and normal line calculator that makes your mathematical explorations even more comprehensive.

How to find the Point Differential Calculator?

The following steps are to be followed for finding the derivative at a point calculator:

You can find our derivative calculator at a point online through search engines like Google or Bing by simply typing relevant keywords like derivative of function at a point calculator. If you're a frequent user of our website, you can also find it on the homepage of our website.

Moreover, this will also let you know about our other derivative tools like partial differentiation calculator and dy/dx calculator.

Benefits of Derivative at a point Calculator

The online tools make the life of the researchers easier by providing them with the facilities of doing fast and accurate calculations. Now the users want to work smarter instead of harder.

Following is the list of some perks of using this derivative of the function at a point calculator:

The derivative point calculator helps to find the derivative of the given function at a certian point.
It makes accurate, reliable calculations without any errors.
The user interface of this online tool is very friendly.
This derivative of the function at a point calculator gives you results in a fraction of a second.
The derivative at-point calculator gives the solution with step-by-step instructions.
This online tool has unlimited access, the user can practice as many examples as he/she can without any premium plan.

So hopefully you will find this tool helpful for you. Moreover, we also offer graph derivative calculator to get the visual analysis of this calculation. Have a happy learning.

Frequently Asked Question

How to Prove a Function is Differentiable at a Point

In dy/dx at a point calculator, To prove that a function is differentiable at a point, we have provided some steps:

Begin with checking the continuity. I.e., if f(x) is non-continuous at x = a then it is undifferentiable.
Determine the derivative of f(x) using the limit definition of the derivative, if it exists then the function is differentiable.

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

Now check the differentiability and check if it exists.

How to Find Derivative at a Point

To find derivative at a point, follow some steps,

If the function has any specific form such as polynomial, trigonometric, etc., it can be differentiated using differentiation rules.
If a function is not straightforward to differentiate then in this scenario limit is used.

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

Let’s see an example of finding a derivative at a point using differentiation:

Differentiate f(x) = x²

$$ f’(x) \;=\; 2x $$

Now we calculate f’ (x) at x = 2

$$ f’(2) \;=\; 2 \times 2 \;=\; 4 $$

What Makes a Function Differentiable at a Point

A function will be differentiable at a point if it satisfies the condition of continuity and existence of the derivative. According to derivative calculator at point, In the continuity condition, the function should be continuous which means it should have no breaks, holes, or jumps.

If it's noncontinuous then it cannot be differentiable.

In the existence of the derivative condition, the derivative of the function must exist which means the limit which is defining the derivative must exist.

What is the Definition of a Derivative at a Point?

The derivative of a point is the measurement of the rate at which the value of the function changes with respect to the independent variable on that specific point. It tells us how fast the function is changing and provides information about the behavior of the function on a point.

It can be presented as,

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

What makes a Point Differentiable

A point will be differentiable if the function itself is differentiable on the point. The derivative of a point calculator explains it a little bit, a function is differentiable at a point if the derivative of the function exists at that point. This means the function should be smooth and have a tangent line.

On the other hand, if the function is differentiable on a specific point then that point is said to be differentiable on that specific point. This means that the function should be smooth at that specific point.

What is the Slope of a Line Parallel to the y-axis?

The slope of a line parallel to the y-axis is vertical which means it doesn’t have a horizontal component to its slope. We may also say that the slope is undefined.

Mathematically, the slope of the line parallel to the y-axis is defined as the change in coordinate divided by the change in the x-coordinate between any two points to evaluate derivative at a point.

However, if a line is vertical, then the change in the x-coordinate is zero as it does not move horizontally. So, dividing by zero results in an undefined slope.

Derivative of a Straight Line Parallel to x-axis is

The derivative of a straight line parallel to the x-axis is a horizontal line. The tilt of this line is zero as it doesn’t rise or fall as it extends horizontally. So the derivative of a horizontal line is zero.

If the function f(x) = c represents a horizontal line then the derivative of this function f’(x) is:

$$ f’(x) \;=\; 0 $$

The reason here is the slope of the horizontal line is equal to zero and constant. Therefore, the derivative represents the rate of change of the function which is also zero for differentiation at a point.

Result:

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