2. What is derivative?
• The derivative of a function of a real variable measures
the sensitivity to change of a quantity (a function or
dependent variable) which is determined by another
quantity (the independent variable).
• For example, the derivative of the position of a moving
object with respect to time is the object's velocity: this
measures how quickly the position of the object changes
when time is advanced.
3. What is differentiation?
• The process of finding a derivative is
• The reverse process is called antidifferentiation.
• The fundamental theorem of calculus states that
antidifferentiation is the same as integration.
• Differentiation is also known as the process to find rate
• Derivative tells us slope of function at any point.
4. As it is also rate of change
• The rate of change of a function is expressed as a ratio
between a change in one variable relative to a
corresponding change in another.
• Rate of change is also given by limit value.
5. Average rate of change
• The Average Rate of Change is defined as
the ratio of the difference in the function f(x)
as it changes from 'a' to 'b' to the difference
between 'a' and 'b'. The average rate of
change is denoted as A(x).
• And is given as the formula
• A(x) =
𝑓 𝑏 −𝑓(𝑎)
• The Average Rate of Change Formula
calculates the slope of a line or a curve on a
6. Average rate of change-Ex
Calculate the average rate of change of a function, f(x) = 3x + 5
as x changes from a to b ?
f(x) = 3x + 5 a = 3 , b = 6
Putting the values
f(3) = 3(3) + 5 :f(6) = 3(6) + 5
f(3)=14 : f(6)=23
The average rate of change is,
𝒇 𝒃 −𝒇 𝒂
A(x) = 3
There are lots of ways to denote the derivative of a function
y = f(x).
f’(x) the derivative of f
the derivative of f with
y’ y prime respect to x.
the derivative of y the derivative of f at x
with respect to x.
20. Chain rule
A special rule, the chain rule, exists for
differentiating a function of another function.
In order to differentiate a function of a function,
y = f(g(x)),
That is to find
, we need to do two things:
1. Substitute u = g(x). This gives us y = f(u)
Next we need to use a formula that is known as the
2. Chain Rule
21. Chain Rule Example
If 𝑦 = 3 2𝑥 − 1 2 𝑢 = 2𝑥 − 1
𝑦 = 3 2𝑥 − 1 2
, 𝑦 = 3 𝑢 2
• Taking derivative to both side
= 6𝑢 -------(i
𝑢 = 2𝑥 − 1
• Taking derivative to both side
24. Higher derivatives. 1st ,2nd 3rd
• Any derivative beyond the first derivative can be referred
to as a higher order derivative.
• The derivative of the function f(x) may be denoted by
• Its double (or "second") derivative is denoted by f ’’(x).
• This is read as "f double prime of x," or "The second
derivative of f(x)."
29. Partial derivatives:
The partial derivative of f is with respect to its
Here ∂ is a rounded d called the partial
derivative symbol. To distinguish it from the
letter d, ∂ is sometimes pronounced "der", "del",
or "partial" instead of "dee"
31. Applications of partial derivatives:
• Derivatives are constantly used in everyday life
to help measure how much something is
changing. They're used by the government in
population censuses, various types of sciences,
and even in economics..
32. Applications of partial derivatives:
• Derivatives in physics.
You can use derivatives a lot in Newton law of motion
where the velocity is defined as the derivative of the
position over time and the acceleration, the derivative of
the velocity over time.
• Derivatives in chemistry.
One use of derivatives in chemistry is when you want to
find the concentration of an element in a product.
33. Concave Up
• The derivative of a function gives the slope
• When the slope continually increases, the function is
• Taking the second derivative actually tells us if the slope
continually increases or decreases.
• When the second derivative is positive, the function is
• f ''(x) > 0 for x > 0;