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4. The Geometrical Concept of the
Derivative

5. Secant Lines, Tangent Lines, and Limit
Definition of a Derivative
A secant line is a straight line joining two
points on a function. It is also equivalent to
the average rate of change, or simply
the gradient or slope between two points.
• Secant line = Average Rate of Change = Slope

6. A tangent line is a straight line that touches a function
at only one point.The tangent line represents
the instantaneous rate of change of the function at
that one point.

7. Let's see what happens as the two points used for the
secant line get closer to one another. Let ∆x represent
the distant between the two points along the x-axis
and determine the limit as ∆x approaches zero.

8. Derivative
• The slope of the tangent line or slope of curve at a point
on the function is equal to the derivative of the function at
the same point

9. • Derivative as the instantaneous rate of
change
The physical Concept of the
Derivative
The derivative tells us the rate of change of one
quantity compared to another at a particular instant
or point (so we call it "instantaneous rate of change").
This concept has many applications in electricity,
dynamics, economics, fluid flow, population
modeling, queuing theory and so on.

11. Instantaneous rate of change
For any given function y = f (x) , the instantaneous rate of
change of the y-values equals the limit of the ratio of change y
to change in x as the change in x approaches zero
we write
Instantaneous change

16. Conclusion
Secant line=Average rate of change=Slope
So the derivative is defined as
=Instantaneous rate of change
=Slope of tangent line
= Slope of curve at a point

17. Example:
What is the slope of the graph of at (4,48)
when x=3.999 , y=47.976003
when x=4.001 , y=48.024003
Thus, the slope is at the point of the graph at which x=4
But, to solve the problem precisely, we compute
=

36. Derivative
• Derivative is a method of calculating rate of
change of one object with respect to another
object.
• Here the objects can be any two comparable
objects, which are obviously related to each
other.
• In mathematics, relation is mostly depicted by
equations.

37. Differential equation
• An equation involving derivative is called
differential equation.

48. Derivative
• Derivative is a door to a part of mathematics, which has been
helpful in understanding the complexities of science in
simpler way.
• It can also be used in real life scenarios,
• For calculating following
• Increase in your height w.r.t your age.
• change in your account balance w.r.t time Or Let's say, change
in finance market w.r.t time.
• GDP change of a country w.r.t the another country.
• Team's performance w.r.t a specific player's performance in
the team.
• Climate change w.r.t increase in pollution.