3.1 The Chain Rule with Tables and Graphs

• The first step in applying the Chain Rule is to find the derivative of the outer function.
• What is the derivative of the inner function $g(x) =$$x^{2}$?


  $2x$
• Steps to apply the Chain Rule

  1️⃣ Identify the outer and inner functions

  2️⃣ Find the derivatives of the outer and inner functions

  3️⃣ Substitute $g(x)$ into $f'(u)$ and multiply by $g'(x)$
• What is the final step in applying the Chain Rule using table values?

  $f'(g(x)) \cdot g'(x)$
• Steps to apply the chain rule using table values

  1️⃣ Identify the composite function $f(g(x))$

  2️⃣ Find the derivatives $f'(u)$ and $g'(x)$ from the table

  3️⃣ Apply the chain rule formula $h'(x) =$$f'(g(x)) \cdot g'(x)$
• What is the inner function in a composite function denoted as $f(g(x))$?


  $g(x)$
• Steps to apply the chain rule using table values

  1️⃣ Identify the composite function

  2️⃣ Find the derivatives of $f'(u)$ and $g'(x)$

  3️⃣ Substitute values into $h'(x) =$$f'(g(x)) \cdot g'(x)$
• The chain rule is used to differentiate composite functions.
• The chain rule formula remains the same regardless of the specific composite function.


  True
• The chain rule formula is h'(x) = f'(g(x)) \cdot g'(x)</latex>.

  True
• What is the derivative of the outer function $f(u) =$$\sin(u)$?


  $f'(u) =$$\cos(u)$
• How do you identify a composite function in a table?

  $g(x)$ values are inputs for $f(x)$.
• When applying the chain rule using table values, $f'(g(1)) =$$f'(2)$.


  True
• How do you find the derivatives $f'(u)$ and $g'(x)$ from graphs?


  Use the slopes of tangents.
• The Chain Rule states that $h'(x) =$$f'(g(x)) \cdot g'(x)$ for $h(x) =$$f(g(x))$.


  True
• In the example $h(x) =$$\sin(x^{2})$, what is the outer function $f(u)$?


  $\sin(u)$
• What is the Chain Rule formula for $h(x) =$$f(g(x))$?


  $h'(x) =$$f'(g(x)) \cdot g'(x)$
• When using the Chain Rule with tables, the derivative of the inner function $g'(x)$ is always found directly from the table.


  False
• The chain rule formula is $h'(x) =$$f'(g(x)) \cdot g'(x)$, where $g'(x)$ is the derivative of the inner function
• In a composite function $f(g(x))$, the output of $g(x)$ is used as the input for f(x)</latex>


  True
• In a table where the output of $g(x)$ is used as the input for $f(x)$, it indicates a composite function.


  True
• The sum rule is used to differentiate non-composite functions.


  True
• What is the outer function in h(x) = \sin(x^{2})</latex>?

  $f(u) =$$\sin(u)$
• The first step in applying the chain rule is to identify the outer and inner functions
• The inner function in $h(x) =$$\sin(x^{2})$ is g(x) = x^{2}</latex>
• Match the step with the corresponding action in applying the chain rule:

  Identify outer and inner functions ↔️ Outer: $f(u) =$$\sin(u)$<br>Inner: $g(x) =$$x^{2}$

  Find derivatives ↔️ $f'(u) =$$\cos(u)$, $g'(x) =$$2x$

  Apply chain rule ↔️ $h'(x) =$$\cos(x^{2}) \cdot 2x$
• What are the two derivatives you need to find when applying the chain rule with table values?

  $f'(u)$ and g'(x)</latex>
• In a graph, if $g(x)$'s output is used as the input for $f(x)$, it illustrates a composite function
• What is a composite function in the context of the Chain Rule?

  One function plugged into another
• Steps to apply the Chain Rule

  1️⃣ Find the derivative of the outer function $f'(u)$

  2️⃣ Substitute $g(x)$ for $u$ in $f'(u)$

  3️⃣ Multiply by the derivative of the inner function $g'(x)$
• Composite functions in tables are identified by checking if the output of one function is used as the input for another.


  True
• In the example $h(x) =$$\sin(x^{2})$, what is the inner function $g(x)$?


  $x^{2}$
• To apply the chain rule, identify the composite function $f(g(x))$ by looking for the output of the inner function being used as the input of the outer function
• When using the chain rule with table values, substitute the values for $f'(u)$ and $g'(x)$ to find the derivative h'(x)
• What does $f'(u)$ represent in the context of the chain rule?


  Derivative of outer function
• How can a composite function be identified in a graph?

  Output of one becomes input of another
• Steps to apply the chain rule

  1️⃣ Identify the outer and inner functions

  2️⃣ Find their derivatives

  3️⃣ Apply the formula $h'(x) =$$f'(g(x)) \cdot g'(x)$
• What is the formula for the chain rule?

  $h'(x) =$$f'(g(x)) \cdot g'(x)$
• What is the outer function in $h(x) =$$\sin(x^{2})$?


  $f(u) =$$\sin(u)$
• The chain rule applied to $h(x) =$$\sin(x^{2})$ results in $h'(x) =$$\cos(x^{2}) \cdot 2x$.


  True