The importance of tangent lines in the application of differentiation is evitable. You can’t get through calculus without them! In math classes, you have learned how to find the slope of a line but never a slope of a curved function. Understanding the concept of the tangent line is necessary as it allows you to find the slope of a curved function at a particular point on the curve.

The word “tangent” means “to touch,” and it’s derived from the Latin word “tangere.” Tangent Line touches the graph of a function at one and only one point. So, to find the tangent line equation, first, you need to find the equation of the curve and then the point at which the tangent is drawn. Point and slope are the only elements you need to find the equation of a tangent line. You will also look at where to find vertical and horizontal tangent lines.

In this article, you will see what a tangent line is, methods to find the tangent line equation, illustrations and how differentiation can be used to find the Tangent line equation.

**Formula**

When y = f (x) is the given curve, the slope of the tangent is, so by point-slope formula, the tangent line equation at (X, Y) is

y – Y = dydx (X). (x – X)

Here (x, y) coordinates on the tangent line, and the derivative is calculated at x = X.

The point-slope formula for a line y – y_{1} = m (x – x_{1}) where (x_{1}, y_{1}) is the point on the line and m is the slope.

**What is a tangent line?**

Consider function f(x) as shown in figure.

The tangent gradient to the given function is the derivative of a function at x = a (say). The tangent drawn at x = a has gradient f ‘ (a)

**How to find the tangent line?**

Step by step calculation

1. Sketch the function and the tangent line

A graph helps the answer to make sense. Sketch the function on paper.

2. Find the first derivative of f (x)

The first derivative of the given function is the equation for the slope of the tangent line.

3. Enter x value of the indicated point into f’ (x)

Find the coordinates of the point and enter the value of x in f’(x) to find the slope of the tangent line.

4. Enter x value into f(x) to find y coordinate.

5. Point-slope form to find Tangent line equation

The point-slope formula for a line y – y_{1} = m (x – x_{1}) where (x_{1}, y_{1}) is the point on the line and m is the slope.

Moreover, if you are asked to find the tangent line equation, you will always find the point where the tangent line interests the graph.

**Example 1**

Find the tangent line equation to f(x) = x^{3} – 3 x^{2 }+ x – 1 at the point x = 3.

f (3) = 3^{3}– 3. 3^{2} + 3 – 1

= 27 – 27 + 3 – 1

= 2

So, the point is (3,2)

To find the gradient,

f’(x) = 3x^{2} – 6 x + 1

f’ (3) = 3. 3^{2} – 6. 3 + 1

= 27 -18 + 1

= 10

To find the tangent line equation

y – y_{1} = m (x – x_{1})

y-2 =10 (x-3)

y-2 =10x-30

y =10x-28

Therefore, this is the required tangent line equation at the point (3,2).

**Example 2**

Find the tangent line equation at x = 5 when g’ (5) = 2 and g (5) = -3.

Slope of the tangent line at x = 5 is g’ (5).

therefore, m = 2 as g’ (5) = 2.

Furthermore, the tangent line contains the point (5, -3).

To find the tangent line equation

y – y_{1} = m (x – x_{1})

y – (-3) = 2 (x – 5)

y + 3 = 2x – 10

y = 2x -13

Therefore, this is the required tangent line equation.

**Example 3**

Find the Tangent line equation of the circle x^{2} + (y – 3)^{2} = 41 through the point (4, -2).

The Centre of the circle is (0,3).

So, yx= 3 – (-2)0 -4 = 5-4

Slope m = 45 as the tangent line is perpendicular.

To write the equation in y = mx + b form, you need to find b, y–intercept.

-2 = 45 (4) + b

-2 = 165 + b

b = -26 5

Therefore, the required tangent line equation is y = 45 x – 265

To solve problems on tangent line equations, all you need to know are the same algebra skills you learned for writing equations of lines. The combination of computing the derivative at the given point and applying the point-slope form of a straight line are the processes involved in finding the tangent lines. Using the first derivative is necessary for solving some problems in calculus.