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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
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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
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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
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Questions & Answers
Explain the following terms .
(1) Abiotic factors in an ecosystem
Abiotic factors are non living components of ecosystem.These include physical and chemical elements like temperature,light,water,soil,air quality and oxygen etc
Qasim
passive process of transport of low-molecular weight material according to its concentration gradient
AI-Robot
what is production?
Catherine
how did the oxygen help a human being
how did the nutrition help the plants
Biology is a branch of Natural science which deals/About living Organism.
evolutionary history and relationship of an organism or group of organisms
AI-Robot
cell is the smallest unit of the humanity biologically
Abraham
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Source:
OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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