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AP Calculus BC
Unit 3: Differentiation: Composite, Implicit, and Inverse Functions
3.1 The Chain Rule with Tables and Graphs
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What is the Chain Rule used to find?
Derivative of a composite function
The Chain Rule is used for differentiating
composite functions
.
True
The Chain Rule states that if
y
=
y =
y
=
f
(
g
(
x
)
)
f(g(x))
f
(
g
(
x
))
, then
d
y
d
x
=
\frac{dy}{dx} =
d
x
d
y
=
d
y
d
g
(
x
)
⋅
d
g
(
x
)
d
x
\frac{dy}{dg(x)} \cdot \frac{dg(x)}{dx}
d
g
(
x
)
d
y
⋅
d
x
d
g
(
x
)
, which is also known as the product
What is the Power Rule used to differentiate?
Power functions
The Sum/Difference Rule is used for
differentiating
sums or differences of functions.
True
What is the correct formula for the Chain Rule?
d
y
d
x
=
\frac{dy}{dx} =
d
x
d
y
=
d
y
d
g
(
x
)
⋅
d
g
(
x
)
d
x
\frac{dy}{dg(x)} \cdot \frac{dg(x)}{dx}
d
g
(
x
)
d
y
⋅
d
x
d
g
(
x
)
To identify a composite function, look for an inner function and an outer
function
What is the inner function of the composite function
f
(
x
)
=
f(x) =
f
(
x
)
=
(
x
2
+
(x^{2} +
(
x
2
+
3
)
4
3)^{4}
3
)
4
?
g
(
x
)
=
g(x) =
g
(
x
)
=
x
2
+
x^{2} +
x
2
+
3
3
3
What are the two key components to look for when identifying a composite function?
Inner and outer function
The outer function in a composite function is the function that wraps around the inner
function
The Chain Rule is used to
differentiate
composite functions.
True
Match the differentiation rule with its definition:
Chain Rule ↔️
d
y
d
x
=
\frac{dy}{dx} =
d
x
d
y
=
d
y
d
g
(
x
)
⋅
d
g
(
x
)
d
x
\frac{dy}{dg(x)} \cdot \frac{dg(x)}{dx}
d
g
(
x
)
d
y
⋅
d
x
d
g
(
x
)
Power Rule ↔️
d
d
x
(
x
n
)
=
\frac{d}{dx}(x^{n}) =
d
x
d
(
x
n
)
=
n
x
n
−
1
nx^{n - 1}
n
x
n
−
1
Constant Rule ↔️
d
d
x
(
c
)
=
\frac{d}{dx}(c) =
d
x
d
(
c
)
=
0
0
0
The inner function
g
(
x
)
g(x)
g
(
x
)
is the function that is "inside" the composite function.
True
In the Chain Rule,
d
y
d
g
(
x
)
\frac{dy}{dg(x)}
d
g
(
x
)
d
y
represents the derivative of the outer function with respect to the inner function
What is a composite function formed by combining two or more functions?
Output becomes input
What is the outer function in a composite function?
Wraps around inner function
In the composite function
f
(
x
)
=
f(x) =
f
(
x
)
=
(
x
2
+
(x^{2} +
(
x
2
+
3
)
4
3)^{4}
3
)
4
, the inner function is x^{2} + 3
A non-composite function is formed by combining other functions.
False
The Constant Rule states that the derivative of a constant is always
zero
Match the differentiation rule with its definition:
Chain Rule ↔️
d
y
d
x
=
\frac{dy}{dx} =
d
x
d
y
=
d
y
d
g
(
x
)
⋅
d
g
(
x
)
d
x
\frac{dy}{dg(x)} \cdot \frac{dg(x)}{dx}
d
g
(
x
)
d
y
⋅
d
x
d
g
(
x
)
Power Rule ↔️
d
d
x
(
x
n
)
=
\frac{d}{dx}(x^{n}) =
d
x
d
(
x
n
)
=
n
x
n
−
1
nx^{n - 1}
n
x
n
−
1
Constant Rule ↔️
d
d
x
(
c
)
=
\frac{d}{dx}(c) =
d
x
d
(
c
)
=
0
0
0
The Chain Rule is specifically designed for differentiating
composite functions
.
True
What is an example of a composite function?
f(x) = (x^{2} + 3)^{4}</latex>
A non-composite function is a function that is not formed by combining other
functions
The function
f
(
x
)
=
f(x) =
f
(
x
)
=
2
x
+
2x +
2
x
+
5
5
5
is a non-composite function.
True
Steps of the Chain Rule process
1️⃣ Identify the inner function
g
(
x
)
g(x)
g
(
x
)
2️⃣ Identify the outer function
f
(
x
)
f(x)
f
(
x
)
3️⃣ Find the derivative of
f
(
g
(
x
)
)
f(g(x))
f
(
g
(
x
))
4️⃣ Find the derivative of
g
(
x
)
g(x)
g
(
x
)
5️⃣ Apply the Chain Rule formula
To identify a composite function, look for an inner function and an outer
function
Steps to apply the Chain Rule
1️⃣ Identify the composite function
f
(
g
(
x
)
)
f(g(x))
f
(
g
(
x
))
2️⃣ Find the derivative of the outer function
d
y
d
g
(
x
)
\frac{dy}{dg(x)}
d
g
(
x
)
d
y
3️⃣ Find the derivative of the inner function
d
g
(
x
)
d
x
\frac{dg(x)}{dx}
d
x
d
g
(
x
)
4️⃣ Multiply the derivatives together
What is the outer function of the composite function
f
(
g
(
x
)
)
f(g(x))
f
(
g
(
x
))
?
f
(
x
)
f(x)
f
(
x
)
Match the differentiation rule with its definition:
Chain Rule ↔️
d
y
d
x
=
\frac{dy}{dx} =
d
x
d
y
=
d
y
d
g
(
x
)
⋅
d
g
(
x
)
d
x
\frac{dy}{dg(x)} \cdot \frac{dg(x)}{dx}
d
g
(
x
)
d
y
⋅
d
x
d
g
(
x
)
Power Rule ↔️
d
d
x
(
x
n
)
=
\frac{d}{dx}(x^{n}) =
d
x
d
(
x
n
)
=
n
x
n
−
1
nx^{n - 1}
n
x
n
−
1
Constant Rule ↔️
d
d
x
(
c
)
=
\frac{d}{dx}(c) =
d
x
d
(
c
)
=
0
0
0
Sum/Difference Rule ↔️
d
d
x
(
f
(
x
)
±
g
(
x
)
)
=
\frac{d}{dx}(f(x) \pm g(x)) =
d
x
d
(
f
(
x
)
±
g
(
x
))
=
d
f
(
x
)
d
x
±
d
g
(
x
)
d
x
\frac{df(x)}{dx} \pm \frac{dg(x)}{dx}
d
x
df
(
x
)
±
d
x
d
g
(
x
)
In a composite function, the inner function is the one that is "inside" the overall
function
The function
f
(
x
)
=
f(x) =
f
(
x
)
=
(
x
2
+
(x^{2} +
(
x
2
+
3
)
4
3)^{4}
3
)
4
is a composite function.
True
What is the outer function in the composite function
f
(
x
)
=
f(x) =
f
(
x
)
=
(
x
2
+
(x^{2} +
(
x
2
+
3
)
4
3)^{4}
3
)
4
?
x
4
x^{4}
x
4
A non-composite function example is
2x + 5
What is the first step in identifying outer and inner functions when applying the Chain Rule?
Identify the composite function
Steps to apply the Chain Rule
1️⃣ Identify the composite function
f
(
g
(
x
)
)
f(g(x))
f
(
g
(
x
))
2️⃣ Determine the outer function
f
(
u
)
f(u)
f
(
u
)
3️⃣ Differentiate the outer function
f
′
(
u
)
f'(u)
f
′
(
u
)
4️⃣ Identify the inner function
g
(
x
)
g(x)
g
(
x
)
5️⃣ Differentiate the inner function
g
′
(
x
)
g'(x)
g
′
(
x
)
6️⃣ Apply the Chain Rule:
d
y
d
x
=
\frac{dy}{dx} =
d
x
d
y
=
f
′
(
g
(
x
)
)
⋅
g
′
(
x
)
f'(g(x)) \cdot g'(x)
f
′
(
g
(
x
))
⋅
g
′
(
x
)
In the example
f
(
g
(
x
)
)
=
f(g(x)) =
f
(
g
(
x
))
=
sin
(
x
2
+
\sin(x^{2} +
sin
(
x
2
+
1
)
1)
1
)
, the outer function is
sin
(
u
)
\sin(u)
sin
(
u
)
What is the inner function in the example
f
(
g
(
x
)
)
=
f(g(x)) =
f
(
g
(
x
))
=
sin
(
x
2
+
\sin(x^{2} +
sin
(
x
2
+
1
)
1)
1
)
?
x
2
+
x^{2} +
x
2
+
1
1
1
What is the final derivative of the composite function
f
(
g
(
x
)
)
=
f(g(x)) =
f
(
g
(
x
))
=
sin
(
x
2
+
\sin(x^{2} +
sin
(
x
2
+
1
)
1)
1
)
after applying the Chain Rule?
2
x
cos
(
x
2
+
2x \cos(x^{2} +
2
x
cos
(
x
2
+
1
)
1)
1
)
When using the Chain Rule with tables, the first step is to identify the
composite
function.
In the example using tables, what is the composite function?
f
(
g
(
x
)
)
=
f(g(x)) =
f
(
g
(
x
))
=
sin
(
x
2
+
\sin(x^{2} +
sin
(
x
2
+
1
)
1)
1
)
See all 114 cards