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?
      ddxf1(x)=\frac{d}{dx}f^{ - 1}(x) =1f(f1(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} +2x1 2x - 1
    • The derivative of y=y =(3x2+ (3x^{2} +2x1)4 2x - 1)^{4} is dydx=\frac{dy}{dx} =4(3x2+ 4(3x^{2} +2x1)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 ddxf1(x)=\frac{d}{dx} f^{ - 1}(x) =1f(f1(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 f1(x)f^{ - 1}(x)?

      f1(x)=f^{ - 1}(x) =x3 \sqrt[3]{x}
    • The derivative of f^{ - 1}(x) = \sqrt[3]{x}</latex> is ddxf1(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} +2x1 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 f1(x)=f^{ - 1}(x) =x3 \sqrt[3]{x} using the inverse derivative formula?

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