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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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Source:
OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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