2.11 The Chain Rule

• The Chain Rule is used to differentiate composite functions.


  True
• Differentiate `y = sin(x^2)` using the Chain Rule.

  `dy/dx = 2x cos(x^2)`
• The outer function in a composite function is the function that the entire expression is inside of
• The Chain Rule formula `dy/dx = dy/du * du/dx` requires differentiating both the outer and inner functions.


  True
• Consider `y = (x^2 + 3)^4`. Identify the outer and inner functions.

  Outer: `u^4`, Inner: `x^2 + 3`
• The Chain Rule is used specifically for differentiating composite functions.
• What is the derivative of `y = sin(x^2)` using the Chain Rule?

  `dy/dx = 2x cos(x^2)`
• In `y = sin(3x^2 + 1)`, the outer function is `f(u) = sin(u)`.

  True
• What rule is used to differentiate a composite function?

  Chain Rule
• The inner function in a composite function is often a more complex expression.


  True
• The inner function in y = sin(3x^2 + 1) is 3x^2 + 1
• After identifying the outer function, the next step is to differentiate it with respect to u
• Find the derivative of y = (x^2 + 1)^3 using the Chain Rule.

  6x(x^2 + 1)^2
• Match the concept with its description:

  Differentiating Outer Function ↔️ Finding the derivative of f(u)

  Chain Rule ↔️ Differentiating the entire composite function
• In the Chain Rule, dy/du represents the derivative of the outer function with respect to the inner
• What is the formula for the Chain Rule when `y = f(u)` and `u = g(x)`?

  `dy/dx = dy/du * du/dx`
• The Product Rule has the formula (uv)' = u'v + uv'
• The Chain Rule applies to functions where one function is nested inside another.


  True
• What is the role of the inner function in a composite function?

  Input to the outer function
• In the Chain Rule, `du/dx` is the derivative of the inner function with respect to x
• Steps to differentiate `y = (x^2 + 3)^4` using the Chain Rule.

  1️⃣ Differentiate the outer function: `dy/du = 4u^3`

  2️⃣ Substitute `u = x^2 + 3`: `dy/du = 4(x^2 + 3)^3`

  3️⃣ Multiply by the derivative of the inner function: `dy/dx = 8x(x^2 + 3)^3`
• The Chain Rule formula remains the same regardless of the complexity of the inner function.

  True
• The outer function in a composite function encloses the entire composite expression
• Match the function type with its definition:

  Outer Function ↔️ Primary function enclosing the composite expression

  Inner Function ↔️ The function inside the outer function
• The outer function in a composite expression is denoted as f(u)
• Identify the outer function in y = sin(3x^2 + 1).

  sin(u)
• What is the first step in differentiating the outer function in the Chain Rule?

  Identify f(u)
• The derivative of the entire composite function dy/dx is obtained by multiplying f'(g(x)) by g'(x).


  True
• Differentiating the outer function involves finding f'(g(x)), which means differentiating f(u) with respect to g(x)
• What is the formula for the Chain Rule?

  $dy / dx =$$dy / du *$$du / dx$
• The Chain Rule requires multiplying the derivatives of the outer and inner functions.


  True
• The final derivative of a composite function in the Chain Rule is found using the formula dy/dx = dy/du * du/dx
• Match the differentiation rule with its formula:

  Chain Rule ↔️ $dy / dx =$$dy / du *$$du / dx$

  Product Rule ↔️ $(uv)' =$$u'v +$$uv'$

  Quotient Rule ↔️ $(u / v)' =$$(u'v - uv') / v^{2}$
• The Chain Rule formula is `dy/dx = dy/du * du/dx
• What does `du/dx` represent in the Chain Rule formula?

  Inner function derivative
• The Product Rule and Quotient Rule are used to differentiate products and quotients of functions, respectively.


  True
• What is the Chain Rule used to differentiate?

  Composite functions
• Steps to identify the outer and inner functions for the Chain Rule:

  1️⃣ Identify the outer function \( f(u) \)

  2️⃣ Identify the inner function \( g(x) \)
• The inner function acts as the input \( u \) for the outer
• What is the formula for differentiating the outer function using the Chain Rule?

  $\frac{dy}{dx} =$$\frac{dy}{du} \cdot \frac{du}{dx}$