2.11 The Chain Rule

• What is the Chain Rule formula used to differentiate composite functions?

  $\frac{d}{dx} f(g(x)) =$$f'(g(x)) \cdot g'(x)$
• What is the derivative of $f(u) =$$u^{2}$?


  $f'(u) =$$2u$
• Apply the Chain Rule to find the derivative of h(x) = (3x + 1)^{2}</latex>.

  $h'(x) =$$18x +$$6$
• The Chain Rule involves differentiating the outer function and then multiplying by the derivative of the inner function.
• The Chain Rule is used to differentiate composite functions.
• What is the first step in applying the Chain Rule?

  Identify inner and outer functions
• Steps to apply the Chain Rule

  1️⃣ Identify the outer function $f(u)$ and inner function $g(x)$

  2️⃣ Differentiate $f(u)$ to find $f'(u)$

  3️⃣ Differentiate $g(x)$ to find $g'(x)$

  4️⃣ Apply the Chain Rule formula $\frac{d}{dx} f(g(x)) =$$f'(g(x)) \cdot g'(x)$
• What is the derivative of $h(x) =$$\sin(2x)$?


  h'(x) = 2\cos(2x)</latex>
• Match the composite function with its outer and inner functions:

  $h(x) =$$\sin(x^{2})$ ↔️ $f(u) =$$\sin(u), g(x) =$$x^{2}$

  $k(x) =$$\sqrt{2x + 1}$ ↔️ $f(u) =$$\sqrt{u}, g(x) =$$2x +$$1$

  $p(x) =$ e^{x^{3} + 5} ↔️ $f(u) =$$e^{u}, g(x) =$$x^{3} +$$5$
• Understanding the outer and inner functions simplifies differentiation using the Chain Rule.
• What is the formula for the Chain Rule?

  $\frac{d}{dx} f(g(x)) =$$f'(g(x)) \cdot g'(x)$
• The Chain Rule states that the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner
• The Chain Rule requires multiplying by the derivative of the inner function.
• What are the two key components of a composite function?

  Outer and inner function
• In a composite function $f(g(x))$, the outer function $f$ acts on the output of the inner
• The inner function operates on the original input $x$.
• What are the steps to apply the Chain Rule?

  Identify, differentiate, apply
• Steps to apply the Chain Rule

  1️⃣ Identify the outer function f(u)</latex> and the inner function $g(x)$

  2️⃣ Differentiate both $f(u)$ and $g(x)$ to find $f'(u)$ and $g'(x)$

  3️⃣ Apply the Chain Rule formula $\frac{d}{dx} f(g(x)) =$$f'(g(x)) \cdot g'(x)$
• What is the process for differentiating nested functions using the Chain Rule?

  Repeated application
• Nested functions require a separate Chain Rule application for each layer of nesting.
• What is the derivative of \sin(x)</latex>?

  $\cos(x)$
• The derivative of $\sin(x)$ is $\cos(x)$, while the derivative of $\cos(x)$ is - \sin(x)
• For h(x) = \sin(2x^{2})</latex>, what is the outer function $f(u)$?


  $\sin(u)$
• The Chain Rule is used to differentiate composite functions of the form f(g(x))
• The Chain Rule formula is $\frac{d}{dx} f(g(x)) =$$f'(g(x)) \cdot g'(x)$.
• Steps to apply the Chain Rule

  1️⃣ Identify the outer function $f(u)$ and the inner function $g(x)$.

  2️⃣ Differentiate $f(u)$ and $g(x)$ to find $f'(u)$ and $g'(x)$.

  3️⃣ Use the Chain Rule formula.
• What is the outer function in $h(x) =$$\sin(x^{2})$?


  $\sin(u)$
• What is the inner function in $h(x) =$$\sin(x^{2})$?


  $x^{2}$
• When applying the Chain Rule, you must differentiate both the outer and inner functions.
• What is the derivative of $h(x) =$$(3x + 1)^{2}$ using the Chain Rule?


  $18x +$$6$
• In the composite function $h(x) =$$\cos(5x)$, the outer function is \cos(u)
• In the composite function h(x) = \cos(5x)</latex>, what is the inner function?

  $5x$
• The Chain Rule formula is \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)
• Match the composite function with its outer and inner functions:

  h(x) = \sqrt{2x + 3}</latex> ↔️ Outer: $\sqrt{u}$, Inner: $2x +$$3$

  $h(x) =$$e^{3x - 1}$ ↔️ Outer: $e^{u}$, Inner: $3x - 1$
• What type of functions does the Chain Rule apply to?

  Composite functions
• The Chain Rule formula is \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)</latex>, where $f(u)$ is the outer function and $g(x)$ is the inner function
• The Chain Rule is used to differentiate composite functions.
• What is the first step in applying the Chain Rule?

  Identify outer and inner functions
• After identifying f(u)</latex> and $g(x)$, the next step is to find their derivatives
• The Chain Rule formula involves multiplying $f'(g(x))$ by $g'(x)$.