x_{0} =

x_{0} =

The tool that we put at your disposal here allows you to find the equation of the tangent line to a curve in a simple and intuitive way. To achieve this, you just need to enter the function of the curve and the value of x0 of the point where you want to find the tangent line. Once these two parameters have been entered, you can press the “Calculate” button to obtain the equation of the tangent line together with its respective graph.

The solution is explained step by step so that the tangent line calculator helps you to study and understand the process to get the tangent line to a function.

Next we explain the theoretical concepts related to the tangent line:

Given a curve whose function is f(x), the tangent line of that function at a given point is the straight line that intersects the curve at that point, thus representing the instantaneous rate of change of the curve. That said, we can establish that the slope of the tangent line at a point of a function is equal to the derivative of the function at the same point. Here is an example:

Formally explained, a straight line is said to be the tangent of a curve y = f (x) at a point x = x0 if the line passes through the point (x0, f (x0)) on the curve and has a slope equal to f'(x0) where f’ is the derivative of f.

The concept of a tangent line is important in calculus because it allows us to study the behavior of a curve at a specific point. For example, we can use the slope of the tangent line to approximate the rate of change of the curve at that point, or to find the equation of the curve at that point.

Starting from the previous definition, we know that the slope of the tangent line is the derivative of the function evaluated at x_{0}.

Using the point-slope equation for the point (x_{0},f(x_{0})) we obtain the formula for the tangent line:

Here are the steps to take to find the equation of a tangent line to a curve at a given point:

- Find the first derivative of f(x).
- Substitute x in f'(x) for the value of x
_{0}at the given point to find the value of the slope. - Substitute x in the original function f(x) for the value of x
_{0}to find value of y at the point where the tangent line is evaluated. - The values obtained in steps 2 and 3 enter them in the point-slope formula, thereby obtaining the equation of the tangent line.
- Plot the results to visually check their validity.

**Example 01: Find the tangent line to f(x)=e ^{x}+1 at x_{0}=3**

**Example 02: Find the tangent line to f(x)=5x ^{3} -3 at x_{0}=1**