Tangent Lines and Normal Lines

Concept Overview

In calculus, the tangent line to a curve at a given point is the line that “just touches” the curve at that point.
The normal line is the line perpendicular to the tangent line at the same point.

Equation of the Tangent Line

For a function y=f(x):

The slope of the tangent line at x=a is the derivative:

$$m_{tangent} = f^\prime(a) $$

The point of tangency is:

(a , f(a))

The equation of the tangent line is:

$$y – f(a) = m_{tangent}(x – a) = f^\prime(a)(x – a)$$

$$y = f^\prime(a)(x – a) + f(a)$$

Equation of the Normal Line

The normal line is perpendicular to the tangent line.
So its slope is the negative reciprocal of the tangent’s slope:

$$m_{normal} = -\frac{1}{f^\prime(a)}$$

Then, the equation of the normal line is:

$$y – f(a) = m_{normal}(x – a) = -\frac{1}{f^\prime(a)}(x – a)$$

$$y = -\frac{1}{f^\prime(a)}(x – a) + f(a)$$

Example

Let’s find the tangent and normal lines to:

$$y = x^2+2x$$ at $$x=1$$

Step 1. Find derivative function$$f^\prime(x)$$

$$f^\prime(x) = 2x +2$$

Step 2. Find slope and tangency point:

$$slope = m_{tangent} = f^\prime(1) = 2(1) + 2 =4$$

$$f(1) = 1^2+2(1) = 3$$ tangency point (1 , 3)

Step 3. Tangent line:

$$y – f(1) = f^\prime(1) (x -1) \longrightarrow y = 4x -1$$​

Step 4. Normal line:

$$y -f(1) = -\frac{1}{f^\prime(1)}(x -1) \longrightarrow y=-\frac{1}{4}x+\frac{13}{4}$$​​

Notes

• The tangent line gives instantaneous rate of change.

• The normal line is useful in geometry and physics (e.g., reflections, perpendicular forces).

• If f′(a)=0, then the tangent is horizontal and the normal is vertical.