The initial concept of lines or surfaces touching each other is acquired from everyday experience. For example, it is intuitively clear that a ruler (has the shape of a straight line) and a hair hoop (has the shape of a circle) located on the table can touch each other. However, in this case, they will have only one common point. Here’s another example: the ball, which is placed on the table, touches its plane. We can say otherwise: the plane of the table touches the ball. In this case, there is also only one joint point for the ball and the plane. In both the first and second considered cases, the connection point is considered the tangency point.

These visual representations need to be transformed into exact mathematical definitions. Read about them below.

Which Line Is a Tangent to the Graph? Lagrange’s Formula 

A direct line, that crosses the point is   image 2021 04 22 15 16 59 considered the tangent line in relation to the graph function f detectable at the determined point . The slope of this line is (consider Figure 1).


The presence of the derivative  at x₀ point indicates the availability of a (non-vertical) tangent at point of the graph of a function characterized by a slope .

The Tangent Line: Derivation of the Equation

It seems appropriate to derive the tangent line equation at a specified point . The direct line equation that has a slope k may be illustrated in the next form: , since

To calculate b, one should pay attention that the tangent is intersecting point A.  Thus,      after transformations we get .

The tangent equation is represented like this:    or

Algorithm for Derivating a Tangent Line Equation

The algorithm for drawing up the mathematical equation for a tangent may be depicted in the following way:

  • Finding a derivative line of the specified function f’ (x);
  • Substituting x₀ meanings into the equation;
  • Substituting x₀ meanings into a derivative;
  • Generating a tangent by applying the formula ;                                                                ;
  • Substituting the argument meanings into the generated tangent equation.

Useful Sample 

Task: There is a graph with the function  It is necessary to determine the tangent at the specified point with the abscissa In the considered case, x₀ = 2, so the function meaning at the selected point is .

The function derivative corresponds to its meaning at the defined point .

Substitute the received meanings into the necessary equation and receive the equation After performing algebraic transformations, we obtain an equation

Therefore, it becomes apparent that the tangent line equation can be represented as  in the considered above case.

How Can a Tangent Line Calculator Help?

A tangent line calculator is a useful mathematics program that automatically searches the tangent equation at a user-specified point x₀. The program displays the equation of a tangent and illustrates the process of completing the task.

This online calculator can be useful for senior pupils of secondary schools in preparation for tests and exams when checking knowledge before the exam, for parents to control the performance of assignments in mathematics and algebra, as well as for college and university students. 

The Benefits of Using the Tangent Line Calculator

Using an online calculator provides a number of advantages, in particular:

  • Quick solution of the task;
  • Finding the right solution;
  • Free online support.

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