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Tangent Line Calculator

Tangent Line Calculator

Select the parabola equation type and write its value along with the point. The calculator will instantly determine the tangent line equation touching at a specified point.


Choose Type:

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.

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

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: \(x = 4y – 3\).

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 Equation Calculator Works?


  • 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


  • 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.

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.


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.