2.11 The Chain Rule

• In Leibniz notation, the Chain Rule is expressed as f'(g(x))
• The Chain Rule involves multiplying the derivatives of the inner and outer functions.


  True
• The Chain Rule has two equivalent forms: Leibniz and Lagrange.


  True
• Both the Leibniz and Lagrange forms of the Chain Rule give the same result
• If $y =$$\sin(x^{2} +$$1)$, then $u =$$x^{2} +$$1$ and \frac{dy}{du} = \cos(u)</latex>. Applying the Chain Rule, the derivative of $y$ with respect to $x$ is 2x \cos(x^{2} + 1)
• If y = (x^{2} + 1)^{3}</latex>, then $\frac{du}{dx} =$$2x$ and $\frac{dy}{du} =$$3u^{2}$


  True
• Match the notation with the corresponding Chain Rule formula:

  Leibniz ↔️ `d/dx[f(g(x))] = f'(g(x)) * g'(x)`

  Lagrange ↔️ `dy/dx = (dy/du) * (du/dx)`
• What is the outer function in the Chain Rule?

  y = f(u)
• What is the Leibniz notation for the Chain Rule?

  $\frac{d}{dx}[f(g(x))] =$$f'(g(x)) \cdot g'(x)$
• The Chain Rule formula in Leibniz notation is \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}</latex>


  True
• If $y =$$(x^{2} +$$1)^{3}$, then $\frac{dy}{du} =$$3u^{2}$. Applying the Chain Rule, the derivative of $y$ with respect to $x$ is 6x(x^{2} + 1)^{2}
• Steps to apply the Chain Rule for composite functions

  1️⃣ Identify the inner function `u = g(x)`

  2️⃣ Identify the outer function `y = f(u)`

  3️⃣ Find the derivatives `du/dx` and `dy/du`

  4️⃣ Apply the Chain Rule formula: `dy/dx = dy/du * du/dx`
• The Chain Rule can be expressed in both Leibniz and Lagrange notations.


  True
• The inner function is denoted as u
• The Chain Rule formula is dy/dx
• Both the Leibniz and Lagrange notations for the Chain Rule yield the same result.


  True
• What is the derivative of $u =$$x^{2} +$$1$?


  $\frac{du}{dx} =$$2x$
• The Chain Rule formula \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}</latex> is also known as the Lagrange notation.
• For $y =$$\sin(x^{2} +$$1)$, the inner function is $u =$$x^{2} +$$1$.


  True
• The Chain Rule is used to differentiate composite functions, which are built by combining two or more functions.
• When differentiating $y =$$e^{x^{2}}$, the inner function is x^{2}.
• Match the Chain Rule notation with its formula:

  Leibniz ↔️ $\frac{d}{dx}[f(g(x))] =$$f'(g(x)) \cdot g'(x)$

  Lagrange ↔️ $\frac{dy}{dx} =$$\left(\frac{dy}{du}\right) \cdot \left(\frac{du}{dx}\right)$
• The Chain Rule has two equivalent forms: Leibniz and Lagrange.


  True
• In a composite function, the inner function $g(x)$ is located within the parentheses
• The first step in applying the Chain Rule is to identify the inner function.

  True
• The derivative of $e^{u}$ with respect to x</latex> is $e^{u} \frac{du}{dx}$.


  True
• What is the inner function in the composite function $y =$$e^{x^{2}}$?


  $u =$$x^{2}$
• The derivative of $\ln u$ with respect to $x$ is $\frac{1}{u} \cdot \frac{du}{dx}$
• What is the first step in solving real-world applications of the Chain Rule?

  Identify the inner and outer functions
• The derivative of a profit function represents the rate of change of profit with respect to the number of units produced.

  True
• The Chain Rule is used to find the derivative of composite functions
• If y = (x^{2} + 1)^{3}</latex>, then $u =$$x^{2} +$$1$ and $f(u) =$$u^{3}$. The derivative of $u$ with respect to $x$ is 2x
• Composite functions are built by combining two or more functions
• Match the notation with the corresponding Chain Rule formula:

  Leibniz ↔️ `d/dx[f(g(x))] = f'(g(x)) * g'(x)`

  Lagrange ↔️ `dy/dx = (dy/du) * (du/dx)`
• When applying the Chain Rule, the first step is to identify the inner function
• Steps to apply the Chain Rule for composite trigonometric functions

  1️⃣ Identify the inner function `u = g(x)`

  2️⃣ Identify the outer function `y = f(u)`

  3️⃣ Find the derivatives `du/dx` and `dy/du`

  4️⃣ Apply the Chain Rule formula: `dy/dx = dy/du * du/dx`
• If $y =$$(x^{2} +$$1)^{3}$, applying the Chain Rule, the derivative of $y$ with respect to $x$ is 6x(x^{2} + 1)^{2}
• What is the first step to apply the Chain Rule?

  Identify the inner function
• To apply the Chain Rule, you need to find the derivatives of both the inner and outer functions.


  True
• The Lagrange notation for the Chain Rule is dy/dx