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**Table of Content**

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.

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.

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.

**Example: **

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

**Solution: **

$$ X_2 = 20y $$

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: \(x = 4y – 3\).

Determine the equation of tangent line at y = 5.

**Solution: **

$$ f (y) = 6 y^2 – 2y + 5f $$

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:

$$ x = 93 + 46y – 184$$

$$ x = 46y – 91$$

- 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

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.

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.