3.1 The Chain Rule with Tables and Graphs

    Cards (160)

    • What does the Chain Rule state for composite functions?
      dydx=\frac{dy}{dx} =f(g(x))g(x) f'(g(x)) \cdot g'(x)
    • The Chain Rule is used when differentiating a function composed of two or more simpler functions.
    • Steps to apply the Chain Rule for y=y =(3x2+ (3x^{2} +5)3 5)^{3}
      1️⃣ Identify outer function f(u)=f(u) =u3 u^{3}
      2️⃣ Identify inner function g(x)=g(x) =3x2+ 3x^{2} +5 5
      3️⃣ Find derivatives f(u)=f'(u) =3u2 3u^{2} and g(x)=g'(x) =6x 6x
      4️⃣ Apply the Chain Rule: dydx=\frac{dy}{dx} =3(3x2+ 3(3x^{2} +5)26x 5)^{2} \cdot 6x
    • The Chain Rule is used to differentiate composite functions
    • What is the Chain Rule formula for y=y =f(g(x)) f(g(x))?

      dydx=\frac{dy}{dx} =f(g(x))g(x) f'(g(x)) \cdot g'(x)
    • The Chain Rule is used when differentiating functions raised to a power.
    • What is the derivative of y=y =(3x2+ (3x^{2} +5)3 5)^{3}?

      dydx=\frac{dy}{dx} =18x(3x2+ 18x(3x^{2} +5)2 5)^{2}
    • What is a composite function formed by nesting one function inside another?
      f(g(x))f(g(x))
    • In a composite function, the function evaluated first is called the inner function.
    • If f(x)=f(x) =x2 x^{2} and g(x)=g(x) =2x+ 2x +3 3, what is f(g(x))</latex>?

      (2x+3)2(2x + 3)^{2}
    • Steps to apply the Chain Rule using tables
      1️⃣ Identify outer and inner functions
      2️⃣ Find derivative values from tables
      3️⃣ Apply the Chain Rule formula: dydx=\frac{dy}{dx} =f(g(x))g(x) f'(g(x)) \cdot g'(x)
      4️⃣ Substitute values into the formula
    • Given tables for f(x)f(x) and g(x)g(x), find dydx\frac{dy}{dx} at x = 2</latex> where y=y =f(g(x)) f(g(x)) and g(2)=g(2) =3 3, g(2)=g'(2) =1 - 1, f(3)=f'(3) =4 4.

      4- 4
    • When applying the Chain Rule, the derivative of the inner function is multiplied by the derivative of the outer function.
    • What is the derivative of y=y =(2x+1)2 (2x + 1)^{2} using the Chain Rule?

      4(2x+1)4(2x + 1)
    • A composite function is created by evaluating one function at the output of another.
    • What is the formula for the Chain Rule?
      dydx=\frac{dy}{dx} =f(g(x))g(x) f'(g(x)) \cdot g'(x)
    • The Chain Rule involves differentiating the outer function evaluated at the inner function and multiplying by the derivative of the inner function.
    • If f(u)=f(u) =u2 u^{2} and g(x) = 2x + 1</latex>, then y=y =(2x+1)2 (2x + 1)^{2}.
    • What is the derivative of f(u)=f(u) =u2 u^{2}?

      f(u)=f'(u) =2u 2u
    • The derivative of g(x)=g(x) =2x+ 2x +1 1 is 2.
    • The derivative of y = (2x + 1)^{2}</latex> is dydx=\frac{dy}{dx} =4(2x+1) 4(2x + 1).
    • What is a composite function formally defined as?
      f(g(x))f(g(x))
    • Match the key components of a composite function with their definitions:
      Inner function ↔️ g(x)g(x)
      Outer function ↔️ f(u)f(u), where u=u =g(x) g(x)
    • If f(x)=f(x) =x2 x^{2} and g(x)=g(x) =2x+ 2x +3 3, then f(g(x)) = (2x + 3)^{2}</latex> is the composite function.
    • When using tables to apply the Chain Rule, you must identify the outer and inner functions.
    • What is the Chain Rule formula when using tables?
      dydx=\frac{dy}{dx} =f(g(x))g(x) f'(g(x)) \cdot g'(x)
    • At x=x =2 2, the value of dydx\frac{dy}{dx} for f(g(x))f(g(x)) is -4.
    • To apply the Chain Rule using graphs, you must interpret the graphs of f(x)f'(x) and g(x)g'(x).
    • What is the second step in applying the Chain Rule using graphs?
      Find g(x)g(x) at a specific xx
    • If g(2)=g(2) =3 3 and g(2)=g'(2) =1 - 1, then f(3)=f'(3) =4 4.
    • The value of dydx\frac{dy}{dx} at x=x =2 2 when using graphs is -4.
    • What is the formula for the Chain Rule?
      dydx=\frac{dy}{dx} =f(g(x))g(x) f'(g(x)) \cdot g'(x)
    • To apply the Chain Rule using graphs, you first use the graphs to find g(x)g(x) and g(x)g'(x) at a specific x
    • The graph of ff' is used to find f'(g(x))</latex> in the Chain Rule.
    • Steps to apply the Chain Rule using graphs:
      1️⃣ Identify the outer and inner functions in the composite function
      2️⃣ Use the graphs to find g(x)g(x) and g(x)g'(x) at a specific xx
      3️⃣ Find f(g(x))f'(g(x)) using the graph of ff'
      4️⃣ Apply the Chain Rule formula
    • What values do you need to find from the graph of gg to apply the Chain Rule at x=x =2 2?

      g(2)g(2) and g(2)g'(2)
    • To find f(g(x))f'(g(x)), you use the graph of ff' and evaluate it at g(x)
    • The Chain Rule requires you to find both g(x)g(x) and g(x)g'(x) using the graph of gg.
    • What does the Chain Rule allow you to differentiate?
      Composite functions
    • Match the step with its description in applying the Chain Rule using graphs:
      Identify outer and inner functions ↔️ Determine the functions in the composite function
      Use graphs to find g(x)g(x) and g(x)g'(x) ↔️ Evaluate the inner function and its derivative
      Find f(g(x))f'(g(x)) using the graph of ff' ↔️ Evaluate the derivative of the outer function at g(x)g(x)
      Apply the Chain Rule ↔️ Multiply f(g(x))f'(g(x)) and g(x)g'(x)
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