Save
AP Calculus AB
Unit 2: Differentiation: Definition and Fundamental Properties
2.5 Applying the Power Rule
Save
Share
Learn
Content
Leaderboard
Share
Learn
Cards (72)
What is the Power Rule used for in differentiation?
Finding derivatives of
x
n
x^{n}
x
n
What is the derivative of
x
3
x^{3}
x
3
?
3x^{2}</latex>
Terms with powers in polynomial expressions involve a variable raised to an
exponent
.
True
To apply the Power Rule, you multiply the original exponent by the base and reduce the exponent by
one
What is the Power Rule used for in calculus?
Finding derivatives
Match each original function with its derivative after applying the Power Rule:
x
3
x^{3}
x
3
↔️
3
x
2
3x^{2}
3
x
2
x
7
x^{7}
x
7
↔️
7
x
6
7x^{6}
7
x
6
x
1
/
2
x^{1 / 2}
x
1/2
↔️
1
2
x
−
1
/
2
\frac{1}{2}x^{ - 1 / 2}
2
1
x
−
1/2
x
−
2
x^{ - 2}
x
−
2
↔️
−
2
x
−
3
- 2x^{ - 3}
−
2
x
−
3
The table illustrates the variable, exponent, and
coefficient
of terms with powers.
The simplified derivative of
x
7
x^{7}
x
7
is
7
x
6
7x^{6}
7
x
6
.
True
Identifying the key components of a term helps in correctly applying the
Power Rule
.
True
The Power Rule can be used to simplify terms with powers in algebraic expressions.
True
What is the derivative of
x
3
x^{3}
x
3
using the Power Rule?
3
x
2
3x^{2}
3
x
2
Steps to combine like terms after applying the Power Rule
1️⃣ Identify terms with the same variable and exponent
2️⃣ Add or subtract the coefficients
3️⃣ Write the combined term
The Power Rule states that if
f
(
x
)
=
f(x) =
f
(
x
)
=
x
n
x^{n}
x
n
, then
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
n
⋅
x
n
−
1
n \cdot x^{n - 1}
n
⋅
x
n
−
1
, where
f
′
(
x
)
f'(x)
f
′
(
x
)
represents the derivative
Match the original function with its derivative:
x
3
x^{3}
x
3
↔️
3
x
2
3x^{2}
3
x
2
x
7
x^{7}
x
7
↔️
7
x
6
7x^{6}
7
x
6
x
1
/
2
x^{1 / 2}
x
1/2
↔️
1
2
x
−
1
/
2
\frac{1}{2}x^{ - 1 / 2}
2
1
x
−
1/2
x
−
2
x^{ - 2}
x
−
2
↔️
−
2
x
−
3
- 2x^{ - 3}
−
2
x
−
3
What is the general formula for the derivative of
x
n
x^{n}
x
n
using the Power Rule?
n
⋅
x
n
−
1
n \cdot x^{n - 1}
n
⋅
x
n
−
1
What is the derivative of
x
7
x^{7}
x
7
?
7
x
6
7x^{6}
7
x
6
What are the key components of a term with a power?
Variable, exponent, coefficient
What is the first step to combine like terms?
Identify like terms
What is the coefficient in the term
2
x
5
2x^{5}
2
x
5
?
2
What is the Power Rule used for?
Differentiation
The term
4
x
4x
4
x
remains unchanged when applying the Power Rule.
True
What does
5
x
2
5x^{2}
5
x
2
become after applying the Power Rule?
10
x
10x
10
x
What is the derivative of
g
(
x
)
=
g(x) =
g
(
x
)
=
1
x
3
+
\frac{1}{x^{3}} +
x
3
1
+
5
x
2
−
x
5x^{2} - \sqrt{x}
5
x
2
−
x
?
g
′
(
x
)
=
g'(x) =
g
′
(
x
)
=
−
3
x
4
+
- \frac{3}{x^{4}} +
−
x
4
3
+
10
x
+
10x +
10
x
+
1
2
x
−
1
/
2
\frac{1}{2}x^{ - 1 / 2}
2
1
x
−
1/2
What is the derivative of
x
\sqrt{x}
x
?
1
2
x
−
1
/
2
\frac{1}{2}x^{ - 1 / 2}
2
1
x
−
1/2
To apply the Power Rule, you multiply the original exponent by the
base
and reduce the exponent by one.
True
If f(x) = x^{4}</latex>, applying the Power Rule,
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
4
x
3
4x^{3}
4
x
3
. This shows the exponent is multiplied by the base
Match the term with its components:
5
x
4
5x^{4}
5
x
4
↔️ Variable: x, Exponent: 4, Coefficient: 5
−
3
x
2
- 3x^{2}
−
3
x
2
↔️ Variable: x, Exponent: 2, Coefficient: -3
2
x
2x
2
x
↔️ Variable: x, Exponent: 1, Coefficient: 2
The derivative of
x
1
/
2
x^{1 / 2}
x
1/2
is
1
2
x
−
1
/
2
\frac{1}{2}x^{ - 1 / 2}
2
1
x
−
1/2
.
True
To apply the Power Rule, you multiply the original exponent by the base and reduce the exponent by one.
True
In the polynomial
f
(
x
)
=
f(x) =
f
(
x
)
=
5
x
4
−
3
x
2
+
5x^{4} - 3x^{2} +
5
x
4
−
3
x
2
+
2
x
−
7
2x - 7
2
x
−
7
, the terms with powers are
5
x
4
5x^{4}
5
x
4
,
−
3
x
2
- 3x^{2}
−
3
x
2
, and
2
x
2x
2
x
.
True
What is the simplified derivative of
x
3
x^{3}
x
3
after applying the Power Rule?
3
x
2
3x^{2}
3
x
2
Match each original term with its simplified derivative:
x
3
x^{3}
x
3
↔️
3
x
2
3x^{2}
3
x
2
x
7
x^{7}
x
7
↔️
7
x
6
7x^{6}
7
x
6
x
1
/
2
x^{1 / 2}
x
1/2
↔️
1
2
x
−
1
/
2
\frac{1}{2}x^{ - 1 / 2}
2
1
x
−
1/2
x
−
2
x^{ - 2}
x
−
2
↔️
−
2
x
−
3
- 2x^{ - 3}
−
2
x
−
3
The derivative of x^{ - 2}</latex> is
-2x^{-3}
The Power Rule is applied to terms with a variable raised to an
exponent
Combining like terms involves adding or subtracting coefficients of terms with the same variable and
exponent
.
True
Combining like terms is necessary to simplify the result after applying the
Power
To apply the Power Rule, you must first multiply the original exponent by the
base
To identify terms with powers, look for variable terms raised to an
exponent
What is the derivative of
x
3
x^{3}
x
3
?
3
x
2
3x^{2}
3
x
2
In the expression
5
x
4
−
3
x
2
+
5x^{4} - 3x^{2} +
5
x
4
−
3
x
2
+
2
x
−
7
2x - 7
2
x
−
7
, the terms with powers are
5
x
4
5x^{4}
5
x
4
,
−
3
x
2
- 3x^{2}
−
3
x
2
, and 2x
See all 72 cards