# Tangent Line Calculator

Tangent Line Calculator is used to determine the equation of a tangent to a given curve. A tangent is a line or a plane that touches a curve or a curved surface at exactly one point.

## What is Tangent Line Calculator?

Tangent Line Calculator is an online tool that helps to find the equation of the tangent line to a given curve when we know the x coordinate of the point of intersection. The point-slope form of a line can be used to find the equation of a tangent. To use the * tangent line calculator*, enter the values in the given input boxes.

### Tangent Line Calculator

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## How to Use Tangent Line Calculator?

Please follow the steps given below to find the equation of the tangent line using the online tangent line calculator:

**Step 1:**Go to Cuemath's online tangent line calculator.**Step 2:**Enter the values in the given input boxes.**Step 3:**Click on the "**Calculate**" button to find the equation of the tangent line.**Step 4:**Click on the "**Reset**" button to clear the fields and enter new values.

## Hoes Does Tangent Line Calculator Work?

To determine the equation of a tangent, we need to know the slope of the line as well as the point where it touches the curve. If we take the first-order derivative of the given function and evaluate it at the point of intersection, we can find the slope of a tangent. Suppose we know the function of the curve, f(x), that the tangent touches and the x coordinate, x_{1}, of the point of intersection. Then we can follow the steps given below to find the equation of the tangent.

- Substitute the value of the x coordinate, x
_{1}, in the given function f(x). This gives us the y coordinate, y_{1}, of the point of intersection. - Differentiate the given function of the curve; f'(x).
- Substitute the value of the x coordinate in f'(x). This will give us the slope of the tangent.
- According to the point-slope form, the equation of a line passing through some point (x
_{0}, y_{0}) with a slope m is given as y - y_{0}= m (x - x_{0}). - Thus, using this concept, the equation of a tangent can be given as y - y
_{1}= f'(x) (x - x_{1}). Substitute the values in this equation to find the tangent line equation.

## Solved Examples on Tangent Line

**Example 1:** Find the equation of the tangent line for the given function f(x) = 3x^{2} at x = 2 and verify it using the online tangent line calculator.

**Solution:**

At x = 2, y = 3x^{2}

= 3 × 2^{2}

= 12

Given: y = f(x) = 3x^{2}

m = f '(x) = 6x

At x = 2

f'(2) = 6 × 2

f'(2) = 12

Equation of tangent line having slope f'(x) = 12 and passing through (2, 12) is

y - y_{1} = f'(x)(x - x_{1})

y - 12 = 12(x - 2)

y - 12 = 12x - 24

12x - y -12 = 0.

Therefore**, **the** **equation of the tangent line is 12x - y - 12 = 0

**Example 2:** Find the equation of the tangent line for the given function f(x) = xln(x) at x = 1 and verify it using the online tangent line calculator.

**Solution:**

At x = 1, y = xln(x)

= 1 × ln(1)

= 0

Given: y = f(x) = xln(x)

m = f '(x) = ln(x) + x / x

f'(x) = lnx + 1

At x = 1,

f'(1) = 0 + 1 = 1

Equation of tangent line having slope f'(x) = 1 and passing through (1, 0) is

y - y_{1} = f'(x)(x - x_{1})

y - 0 = 1(x - 1)

x - y - 1 = 0

Therefore**, **the** **equation of the tangent line is x - y - 1 = 0

Similarly, you can use the tangent line calculator to find the equation of the tangent line for the following:

- y = e
^{x}ln(x) at x = 1. - y = 5x
^{3}+ 1.2x at x = 3.

**☛ Math Calculators:**

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