Tangent line is a line that just touches a curve at one point without cutting across it. It can be a line through a pair of infinitely close points on the curve. Formally it is a line which intersects a differentiable curve at a point where the slope of the curve is equal to the slope of the line.

The word tangent comes from the latin word tangere means to touch. First definition of tangent line was 'a right line which touches a curve, but which when produced, does not cut it'. Tangent line just touches the surface of the curve at a point and the point at which it touches the curve is called the point of tangency.
Tangent Line

A line tangent to a curve has two properties.
1. The line shares a point with the curve.
2. Derivative of the curve is equal to the slope of the line.
In calculus, tangent line is a straight line to a given curve y = f(x) at a point $x_{0}$ which passes through the point ($x_{0},f(x_{0})$) on the curve and its slope is given by f'($x_{0}$)

where f'(x) is the derivative of the function f(x).

For most points the tangent touches the curve without crossing it. A point where the tangent crosses the curve is known as the inflection point. Graph of a cubic function has inflection point however, circles, ellipses, parabolas and hyperbolas do not have an inflection point.

Tangent Line of Curve
Equation of a tangent line can be found by using the slope intercept formula for a line or by using point slope formula for a line.

Slope intercept formula for a line

The formula is y = mx + b
where m : slope of the line
b : y-intercept

Slope point formula for a line

The formula is y - $y_{1}$ = m (x - $x_{1}$)
Point on the line is denoted by ($x_{1}$, $y_{1}$)
m : slope of the line.
Given below are the simple steps to find the equation of a tangent line.
1. Find the derivative of the given function.
2. From the given points find the slope of the tangent.
3. Using the slope point form, find the equation of the tangent line.
Tangent line to a circle is a line through a pair of infinitely close points on the circle. A tangent touches a circle in exactly one place so the point will be perpendicular to the radius. Tangent intersects the circle's radius at 90$^{\circ}$ angle.

If two tangents form a common point then they are said to be congruent. Some lines do not intersect the circle at all where as secant lines intersect the circle at two points. In the interior of a circle no tangent line can be drawn and a line tangent to a circle is perpendicular to the radius to the point of tangency.Tangent Line of a Circle

Solved Examples

Question 1: Find the equation of the tangent line for the curve f(x) = x$^{4}$ + 10 at x = 3.
Solution:
 
Given f(x) = x$^{4}$ + 10
f'(x) = 4x$^{3}$

At x = 3, slope of the tangent line is f'(3)
f'(3) = 4(3)$^{3}$
f'(3) = 108 (slope)

As the value of $x_{1}$ is known, the value of $y_{1}$ is found by plugging x = 3 in f(x)

f(3) = 3$^{4}$ + 10
f(3) = 91

Using the slope point form, equation of a tangent line can now be found.
At  the point (3, 91), f'(3) = 108 equation of tangent line is
y - $y_{1}$ = m (x - $x_{1}$)
y - 91 = 108 (x - 3)
y - 91 = 108x - 324
y = 108 x - 233
 

Question 2: Find the equation of the tangent line for the curve f(x) = 2x$^{3}$  + 5x$^{2}$ + 7x + 3 at the point (1, -1 )

Solution:
 
Given f(x) = 2x$^{3}$  + 5x$^{2}$ + 7x + 3
f'(x) = 6x$^{2}$ + 10x +7
Find the slope at x = 1
f'(1) = 6 + 10 + 7 = 23 (slope)

As the point ($x_{1}, y_{1}$) is given, using the slope point form equation of a tangent line can now be found.
At the point (1, -1), f'(1) = 23 equation of tangent line is
y - $y_{1}$ = m (x - $x_{1}$)
y  + 1 = 23 (x - 1)
y + 1 = 23 x - 23
y = 23 x - 24