3.5 Selecting Procedures for Calculating Derivatives

Cards (35)

  • What is the first step in calculating derivatives for different types of functions?
    Identify the function type
  • A composite function is formed by composing two or more functions
  • An implicit function has y explicitly isolated in the equation.
    False
  • What type of function reverses the input and output of another function?
    Inverse function
  • Match the function type with its differentiation procedure:
    Composite Function ↔️ Chain Rule
    Implicit Function ↔️ Implicit Differentiation
    Inverse Function ↔️ Inverse Function Rule
  • The Chain Rule is used to differentiate composite functions
  • Matching the function type to the correct differentiation procedure is crucial for accuracy.
  • What is the formula for the Chain Rule if y = f(g(x))</latex>?
    dydx=\frac{dy}{dx} =f′(g(x))⋅g′(x) f'(g(x)) \cdot g'(x)
  • Steps to perform implicit differentiation for x2+x^{2} +y2= y^{2} =1 1.

    1️⃣ Differentiate both sides with respect to xx
    2️⃣ Apply the chain rule when differentiating y2y^{2}
    3️⃣ Solve for dydx\frac{dy}{dx}
  • What is the derivative of f(u)=f(u) =u4 u^{4}?

    f′(u)=f'(u) =4u3 4u^{3}
  • The chain rule in implicit differentiation is used when differentiating y2y^{2} with respect to xx
  • What is dydx\frac{dy}{dx} for the implicit function x2+x^{2} +y2= y^{2} =1 1?

    dydx=\frac{dy}{dx} =−xy - \frac{x}{y}
  • Match the function type with its differentiation procedure:
    Composite Function ↔️ Chain Rule
    Implicit Function ↔️ Implicit Differentiation
    Inverse Function ↔️ Inverse Function Rule
  • Composite functions are differentiated using the chain rule.
  • Steps to differentiate a composite function using the chain rule:
    1️⃣ Identify the outer and inner functions
    2️⃣ Find the derivative of the outer function
    3️⃣ Find the derivative of the inner function
    4️⃣ Multiply the derivatives
  • To differentiate an implicit function, you first differentiate both sides with respect to x
  • What is the inverse function rule for differentiation?
    ddxf−1(x)=\frac{d}{dx}f^{ - 1}(x) =1f′(f−1(x)) \frac{1}{f'(f^{ - 1}(x))}
  • What is the key procedure in implicit differentiation?
    Solve for dydx\frac{dy}{dx}
  • The inverse function rule states that \frac{d}{dx}f^{ - 1}(x) = \frac{1}{f'(f^{ - 1}(x))}</latex>, where f′(x)f'(x) is the derivative of the original function
  • 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 inverse function rule is used to find the derivative of the inverse of a function.
  • The chain rule is used to differentiate composite functions.
  • Steps to apply the chain rule
    1️⃣ Identify the outer function f(u)f(u) and inner function g(x)g(x)
    2️⃣ Find their derivatives: f′(u)f'(u) and g′(x)g'(x)
    3️⃣ Apply the chain rule formula
  • What are the outer and inner functions for y = (3x^{2} + 2x - 1)^{4}</latex>?
    f(u)=f(u) =u4,g(x)= u^{4}, g(x) =3x2+ 3x^{2} +2x−1 2x - 1
  • The derivative of y=y =(3x2+ (3x^{2} +2x−1)4 2x - 1)^{4} is dydx=\frac{dy}{dx} =4(3x2+ 4(3x^{2} +2x−1)3⋅(6x+2) 2x - 1)^{3} \cdot (6x + 2), using the chain rule.
  • Implicit differentiation involves differentiating both sides of an equation with respect to xx.
  • Steps to perform implicit differentiation
    1️⃣ Differentiate both sides with respect to xx
    2️⃣ Apply chain rule where necessary
    3️⃣ Solve for dydx\frac{dy}{dx}
  • What is the derivative dydx\frac{dy}{dx} for the equation x^{2} + y^{2} = 1</latex>?

    dydx=\frac{dy}{dx} =−xy - \frac{x}{y}
  • The inverse derivative formula states that ddxf−1(x)=\frac{d}{dx} f^{ - 1}(x) =1f′(f−1(x)) \frac{1}{f'(f^{ - 1}(x))}, where f′(x)f'(x) is the derivative of the original function.
  • If f(x)=f(x) =x3 x^{3}, what is f−1(x)f^{ - 1}(x)?

    f−1(x)=f^{ - 1}(x) =x3 \sqrt[3]{x}
  • The derivative of f^{ - 1}(x) = \sqrt[3]{x}</latex> is ddxf−1(x)=\frac{d}{dx} f^{ - 1}(x) =13x23 \frac{1}{3x^{\frac{2}{3}}}, using the inverse derivative formula.
  • Which two differentiation procedures are often combined for complex functions?
    Chain rule and implicit differentiation
  • The chain rule is frequently used within implicit differentiation and the inverse function rule.
  • Steps to apply the chain rule for y = (3x^{2} + 2x - 1)^{4}</latex>
    1️⃣ Identify the outer function f(u)=f(u) =u4 u^{4} and inner function g(x)=g(x) =3x2+ 3x^{2} +2x−1 2x - 1
    2️⃣ Find their derivatives: f′(u)=f'(u) =4u3 4u^{3} and g′(x)=g'(x) =6x+ 6x +2 2
    3️⃣ Apply the chain rule formula
  • What is the derivative of f−1(x)=f^{ - 1}(x) =x3 \sqrt[3]{x} using the inverse derivative formula?

    13x23\frac{1}{3x^{\frac{2}{3}}}