The term “tangent” means “to touch”. The Latin word for the same is “tangere”. In general, we can say that the line that intersects the circle exactly at one point on its circumference and never enters the circle’s interior is a tangent. A circle can have many tangents. They are perpendicular to the radius.

Derivatives are commonly used in mathematics to determine how the value of one variable changes in relation to another. Derivatives are seen and debated in real life as well. For example, the word speed is frequently used, but few people realise that speed is actually a derivative; speed is the change in distance vs. time. Let’s take a look at some of the most common derivative uses.

• To determine a function’s rate of change with regard to variables.
• Curves that are tangent and normal are found.
• To determine a function’s maximum and minimum values.
• To determine if a function is increasing or decreasing, as well as the interval in which it is increasing or decreasing.
• To determine the concavity of a function, whether the function is concave up or down.
• To compute velocity and acceleration from velocity, use the formulas below (velocity is a change of displacement with respect to time and acceleration is the change of velocity with respect to time).
• To determine the approximate value of a number of variables (for example square root, cube root of a number).

Tangent

A tangent is a line formed from an external point that goes through a point on a curve in geometry. When riding a bicycle, every point on the circumference of the wheel forms a tangent with the road.

Point of Contact

The single point of junction where the straight-line contacts or intersects the circle is known as the point of tangency. The point of tangency is shown by point P in the diagram above.

Properties of the Tangent to the curve.

The Tangent has following characteristics:

• Only one point on a curve is touched by a tangent.
• A tangent is a line that never passes through the centre of the circle.
• At the point of tangency, the tangent makes a right-angle contact with the radius of the circle.

Normal

A normal line to a point (x,y) on a curve is perpendicular to the tangent line and passes through the point (x,y). The slopes of the normal and tangent lines will be opposite reciprocals of each other since they are perpendicular. Take the derivative of the function at the point (x,y) to obtain the slope of the tangent line.

Equation of the Tangent and Equation of the Normal to the curve.

The Equation of the Tangent and the equation of the Normal a curve whose equation is given below;

y = f(x)

Let us take a point At “a” point at which The Equation of the Tangent and the equation of the Normal curve are drawn. 

So, the equation of tangent in point-slope form is 

{y – f(a) } / (x – a) = f'(a),

 and using equation  mTangent × mNormal = -1,[Product of the slope of tangent and normal is -1]

so the Equation of normal is: (y – f(a))/(x – a) = -1/f'(a).

Point to Remember:

• A tangent line is a straight line that only meets a curve once.
• A slope is similar to a derivative in that it describes how much a function changes at a given point.
• A derivative, on the other hand, might be confused with h slope; it’s a calculus technique that allows you to compute the slope of a function.
• Perpendicular lines are formed when two crossing lines produce four right angles: these four right angles are called perpendicular.
• The equation of a line is usually stated as y = mx + b, where m is the slope, b is the y-intercept, and (x,y) represents any point on the line.