Home
Precalculus Introduction to calculus Derivatives
A General Note
Notations for the derivative
The equation of the derivative of a function
f
(
x
)
is written as
y
′
=
f
′
(
x
)
, where
y
=
f
(
x
)
.
The notation
f
′
(
x
)
is read as “
f
prime of
x
. ” Alternate notations for the derivative include the following:
f
′
(
x
)
=
y
′
=
d
y
d
x
=
d
f
d
x
=
d
d
x
f
(
x
)
=
D
f
(
x
)
The expression
f
′
(
x
)
is now a function of
x ; this function gives the slope of the curve
y
=
f
(
x
)
at any value of
x
.
The derivative of a function
f
(
x
)
at a point
x
=
a
is denoted
f
′
(
a
)
.
How To
Given a function
f
, find the derivative by applying the definition of the derivative.
Calculate
f
(
a
+
h
)
.
Calculate
f
(
a
)
.
Substitute and simplify
f
(
a
+
h
)
−
f
(
a
)
h
.
Evaluate the limit if it exists:
f
′
(
a
)
=
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
.
Finding the derivative of a polynomial function
Find the derivative of the function
f
(
x
)
=
x
2
−
3
x
+
5
at
x
=
a
.
We have:
f
′
(
a
)
=
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
Definition of a derivative
Substitute
f
(
a
+
h
)
=
(
a
+
h
)
2
−
3
(
a
+
h
)
+
5
and
f
(
a
)
=
a
2
−
3
a
+
5.
f
′
(
a
)
=
lim
h
→
0
(
a
+
h
)
(
a
+
h
)
−
3
(
a
+
h
)
+
5
−
(
a
2
−
3
a
+
5
)
h
=
lim
h
→
0
a
2
+
2
a
h
+
h
2
−
3
a
−
3
h
+
5
−
a
2
+
3
a
−
5
h
Evaluate to remove parentheses
.
=
lim
h
→
0
a
2
+
2
a
h
+
h
2
−
3
a
−
3
h
+
5
−
a
2
+
3
a
−
5
h
Simplify
.
=
lim
h
→
0
2
a
h
+
h
2
−
3
h
h
=
lim
h
→
0
h
(
2
a
+
h
−
3
)
h
Factor out an
h
.
=
2
a
+
0
−
3
Evaluate the limit
.
=
2
a
−
3
Got questions? Get instant answers now! Got questions? Get instant answers now!
Finding derivatives of rational functions
To find the derivative of a rational function, we will sometimes simplify the expression using algebraic techniques we have already learned.
Finding the derivative of a rational function
Find the derivative of the function
f
(
x
)
=
3
+
x
2
−
x
at
x
=
a
.
f
′
(
a
)
=
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
=
lim
h
→
0
3
+
(
a
+
h
)
2
−
(
a
+
h
)
−
(
3
+
a
2
−
a
)
h
Substitute
f
(
a
+
h
)
and
f
(
a
)
=
lim
h
→
0
(
2
−
(
a
+
h
)
)
(
2
−
a
)
[
3
+
(
a
+
h
)
2
−
(
a
+
h
)
−
(
3
+
a
2
−
a
)
]
(
2
−
(
a
+
h
)
)
(
2
−
a
)
(
h
)
Multiply numerator and denominator by
(
2
−
(
a
+
h
)
)
(
2
−
a
)
=
lim
h
→
0
(
2
−
(
a
+
h
)
)
(
2
−
a
)
(
3
+
(
a
+
h
)
(
2
−
(
a
+
h
)
)
)
−
(
2
−
(
a
+
h
)
)
(
2
−
a
)
(
3
+
a
2
−
a
)
(
2
−
(
a
+
h
)
)
(
2
−
a
)
(
h
)
Distribute
=
lim
h
→
0
6
−
3
a
+
2
a
−
a
2
+
2
h
−
a
h
−
6
+
3
a
+
3
h
−
2
a
+
a
2
+
a
h
(
2
−
(
a
+
h
)
)
(
2
−
a
)
(
h
)
Multiply
=
lim
h
→
0
5
h
(
2
−
(
a
+
h
)
)
(
2
−
a
)
(
h
)
Combine like terms
=
lim
h
→
0
5
(
2
−
(
a
+
h
)
)
(
2
−
a
)
Cancel like factors
=
5
(
2
−
(
a
+
0
)
)
(
2
−
a
)
=
5
(
2
−
a
)
(
2
−
a
)
=
5
(
2
−
a
)
2
Evaluate the limit
Got questions? Get instant answers now! Got questions? Get instant answers now!
Finding derivatives of functions with roots
To find derivatives of functions with roots, we use the methods we have learned to find limits of functions with roots, including multiplying by a conjugate.
Finding the derivative of a function with a root
Find the
derivative
f
(
x
)
=
4
x
at
x
=
36.
We have
f
′
(
a
)
=
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
=
lim
h
→
0
4
a
+
h
−
4
a
h
Substitute
f
(
a
+
h
)
and
f
(
a
)
Multiply the numerator and denominator by the conjugate:
4
a
+
h
+
4
a
4
a
+
h
+
4
a
.
f
′
(
a
)
=
lim
h
→
0
(
4
a
+
h
−
4
a
h
)
⋅
(
4
a
+
h
+
4
a
4
a
+
h
+
4
a
)
=
lim
h
→
0
(
16
(
a
+
h
)
−
16
a
h
4
(
a
+
h
+
4
a
)
)
Multiply
.
=
lim
h
→
0
(
16
a
+
16
h
−
16
a
h
4
(
a
+
h
+
4
a
)
)
Distribute and combine like terms
.
=
lim
h
→
0
(
16
h
h
(
4
a
+
h
+
4
a
)
)
Simplify
.
=
lim
h
→
0
(
16
4
a
+
h
+
4
a
)
Evaluate the limit by letting
h
=
0.
=
16
8
a
=
2
a
f
′
(
36
)
=
2
36
Evaluate the derivative at
x
=
36.
=
2
6
=
1
3
Got questions? Get instant answers now! Got questions? Get instant answers now!
Questions & Answers
start new n questions too
summarize halerambos & holbon
the Three stages of Auguste Comte
what are agents of socialization
definition of sociology of education
Nuhu
definition of sociology of education
Emmaunella
shared beliefs, values, and practices
AI-Robot
What are the two type of scientific method
what are the scientific method of sociology
Man
the process wherein people come to understand societal norms and expectations, to accept society's beliefs, and to be aware of societal values
AI-Robot
scientific method in doing research
ogunniran
defimition of sickness in afica
Anita
list and explain the terms that found in society
list and explain the terms that found in society
Mukhtar
what are the agents of socialization
Antonio
Family
Peer group
Institution
Abdulwajud
I mean the definition
Antonio
ways of perceived deviance indifferent society
reasons of joining groups
SAM
to bring development to the nation at large
Hyellafiya
entails of consultative and consensus building from others
Gadama
World first Sociologist?
Abu
What is evolutionary model
Evolution models refer to mathematical and computational representations of the processes involved in biological evolution. These models aim to simulate and understand how species change over time through mechanisms such as natural selection, genetic drift, and mutation. Evolutionary models can be u
faruk
what are the modern trends in religious behaviours
shared standards of acceptable behavior by the group or appropriate behavior in a particular institution or those behaviors that are acceptable in a society
Lucius
that is how i understood it
Lucius
examples of societal norms
Diamond
Discuss the characteristics of the research located within positivist and the interpretivist paradigm
Got questions? Join the online conversation and get instant answers!
Source:
OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.
