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Tangent Line Calculator

Tangent Line Calculator

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Enter a function: f(x)


Enter a point: (x₀)

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An online tangent line calculator will help you to determine the tangent line to the implicit, parametric, polar, and explicit at a particular point. Apart from this, the equation of tangent line calculator can find the horizontal and vertical tangent lines as well. So, keep reading to understand how to find tangent line and slope of a tangent line with the help of tangent line equation.

Let’s dive in!

What is a Tangent Line?

The line and the curve intersect at a point, that point is called tangent point. So, a tangent is a line that just touches the curve at a point. The point where a line and a curve meet is called the point of tangency.

Tangent Line Calculator

Therefore with this tangent line calculator, you will be able to calculate the slope of tangent line. However, an online Point Slope Form Calculator will find the equation of a line by using two coordinate points and the slope of the line.

Tangent Line Formula:

Well, there are various variables used to determine the equation of the tangent line to the curve at a particular point:

  • The slope of a tangent line
  • On the curve, where the tangent line is passing

So the Standard equation of tangent line:

$$ y – y_1 = (m)(x – x_1)$$

Where (x_1 and y_1) are the line coordinate points and “m” is the slope of the line.


Find the tangent equation to the parabola x_2 = 20y at the point (2, -4):


$$ X_2 = 20y $$

Differentiate with respect to “y”:

$$ 2x (dx/dy) = 20 (1)$$

$$ m = dx / dy = 20/2x ==> 5/x $$

So, slope at the point (2, -4):

$$ m = 4 / (-4) ==> -1 $$

Equation of Tangent line is:

$$ (x – x_1) = m (y – y_1) $$

$$ (x – (-4)) = (-1) (y – 2) $$

$$ x + 4 = -y + 2 $$

$$ y + x – 2 + 4 = 0 $$

$$ y + x + 2 = 0 $$

When using slope of tangent line calculator, the slope intercepts formula for a line is:

$$ x = my + b $$

Where “m” slope of the line and “b” is the x intercept.

So, the results will be:

$$ x = 4 y^2 – 4y + 1 at y = 1$$

Result = 4

Therefore, if you input the curve “x= 4y^2 – 4y + 1” into our online calculator, you will get the equation of the tangent: \(x = 4y – 3\).

Moreover, an Online Derivative Calculator helps to find the derivative of the function with respect to a given variable and shows you the step-by-step differentiation.

Determining the Equation of a Tangent Line at a Point:

Determine the equation of tangent line at y = 5.


$$ f (y) = 6 y^2 – 2y + 5f $$

First of all, substitute y = 5 into the function:

$$ f (5) = 6 (5)^2 – 2 (5) + 5 $$

$$ f (5) = 150 – 10 + 5 ==> f (5) = 165$$

by taking the derivative and plug in y = 5:

$$ f ‘ (y) = 12y – 2 $$

$$ f ‘(5) = 12 (5) – 2 $$

$$ f ‘ (5) = 58 $$

Then, add both f (5) and f'(5) into the equation of a tangent line, along with 5 for a:

$$y = 93 + 46 (y – 5)$$

so the result will be:

$$ x = 93 + 46y – 184$$

$$ x = 46y – 91$$

How Tangent Line Calculator Works?

An online tangent line equation calculator gives the slope and the equation of a tangent line by following these steps:


  • Firstly, choose the type of curve either explicit, parametric, or polar from the drop-down list.
  • Now, Enter the values of the function
  • Then, enter a particular point where you want to find a tangent line
  • Click the calculate


Equation of tangent line calculator will display:

  • Your input and answer 
  • Then find the function and take the derivative of a certain function
  • Lastly, the calculator determines the slope and the tangent line


Why should we Search Tangent of Function Graphs?

To find a tangent to a graph in a point, we can say that a certain graph has the same slope as a tangent. Then use the tangent to indicate the slope of the graph.

Discuss Two Tangent Properties.

Is Slope of a Tangent Line the Derivative?

The derivative of a function gives the slope of a line tangent to the function at some point on the graph. This will be used to find the equation of a tangent line.

What is the Different Between Slope and Tangent?

The slope of the tangent is the gradient of a particular line; the tangent to a curve at a point is a straight line touching the curve at a point.

The Takeaway:

Use this handy tangent line calculator to find the tangent line to the several curves at the given point with a complete solution. Therefore, students and teachers can perform all these calculations manually. However, this is a difficult and time-consuming task. By using an online tangent line equation calculator you can determine tangent lines seamlessly at specific points numerous times.


From the source of Wikipedia: Tangent line to a curve, Analytical approach, Intuitive description.

From the source of Krista King: What Is The Tangent Line, the tangent line at a particular point, Equation Of The Tangent Line.

From the source of Paul Notes: Tangent Lines And Rates Of Change, Velocity Problem, Change of Notation.