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AP Calculus AB
Unit 2: Differentiation: Definition and Fundamental Properties
2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
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The Constant Rule states that if f(x) = c, where c is a constant, then f'(x) =
0
What is the derivative of the difference of two functions according to the Difference Rule?
Difference of derivatives
What is the derivative of a constant multiple of a function according to the Constant Multiple Rule?
Constant multiple of derivative
What is the formula for the Constant Multiple Rule?
(kf(x))' = kf'(x)</latex>
What is the derivative of the sum of two functions according to the Sum Rule?
Sum of derivatives
What is the derivative of the difference of two functions according to the Difference Rule?
Difference of derivatives
If f(x) = x^3 and g(x) = 2x, what is (f - g)'(x) according to the Difference Rule?
3x^2 - 2
The Difference Rule is similar to the
Sum Rule
, but uses subtraction instead of addition.
True
The derivative of a
constant
is always zero.
True
The Constant Rule states that the derivative of a constant is zero.
True
The Difference Rule states that the derivative of the difference of two functions is the difference of their
individual
derivatives.
True
If f(x) = 5x^2, what is (2f(x))'?
20x
What is the derivative of any constant function according to the Constant Rule?
Zero
What is the derivative of the sum of two functions according to the Sum Rule?
Sum of derivatives
If f(x) = x^2 and g(x) = 3x, then (f + g)'(x) = 2x + 3 according to the Sum Rule.
True
If f(x) = x^3 and g(x) = 2x, then (f - g)'(x) = 3x^2 - 2 according to the
Difference Rule
.
True
If f(x) = 5, then f'(x) = 0 according to the Constant Rule.
True
The Sum Rule states that if (f + g)'(x) = f'(x) + g'(x), then the operation being performed is
sum
The Difference Rule states that if (f - g)'(x) = f'(x) - g'(x), then the operation being performed is
difference
The Constant Multiple Rule states that if (kf(x))' = kf'(x), then the constant k is being multiplied by the derivative of
f(x)
If f(x) = 2x^3, then f'(x) = 6x^2. Using the Constant Multiple Rule, we have (3f(x))' =
18x^2
The Sum Rule states that if (f + g)'(x) = f'(x) + g'(x), then the operation being performed is
sum
The Difference Rule states that if (f - g)'(x) = f'(x) - g'(x), then the operation being performed is
difference
What does the Difference Rule state for the derivative of (f - g)'(x)?
f'(x) - g'(x)
What does the Constant Multiple Rule state for the derivative of (kf(x))'?
kf'(x)
What is the derivative of f(x) = 5?
0
What does the Sum Rule state for the derivative of (f + g)'(x)?
f'(x) + g'(x)
What is (f - g)'(x) if f(x) = x^3 and g(x) = 2x?
3x^2 - 2
What does the Constant Multiple Rule state about the derivative of a constant multiple of a function?
Constant multiple of derivative
What does the Sum Rule state about the derivative of the sum of two functions?
Sum of their derivatives
The derivative of a constant function is always
zero
The Constant Rule applies when a function is constant and its
derivative
is 0.
True
What is the formula for the Constant Multiple Rule?
(
k
f
(
x
)
)
′
=
(kf(x))' =
(
k
f
(
x
)
)
′
=
k
f
′
(
x
)
kf'(x)
k
f
′
(
x
)
Mathematically, the Constant Multiple Rule is
(kf(x))' = kf'(x)
The Constant Multiple Rule contrasts with the Constant Rule, which results in a
derivative
of 0.
True
If
f
(
x
)
=
f(x) =
f
(
x
)
=
5
5
5
, what is
f
′
(
x
)
f'(x)
f
′
(
x
)
?
0
Mathematically, the Sum Rule is
(f + g)'(x) = f'(x) + g'(x)
If
f
(
x
)
=
f(x) =
f
(
x
)
=
x
3
x^{3}
x
3
and
g
(
x
)
=
g(x) =
g
(
x
)
=
2
x
2x
2
x
, what is (f - g)'(x)</latex>?
3
x
2
−
2
3x^{2} - 2
3
x
2
−
2
If
f
(
x
)
=
f(x) =
f
(
x
)
=
2
x
3
2x^{3}
2
x
3
and
f
′
(
x
)
=
f'(x) =
f
′
(
x
)
=
6
x
2
6x^{2}
6
x
2
, then
(
3
f
(
x
)
)
′
=
(3f(x))' =
(
3
f
(
x
)
)
′
=
18
x
2
18x^{2}
18
x
2
by applying the Constant Multiple Rule.18x^{2}
The derivative of a constant is always
zero
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