2.9 The Quotient Rule

• u'(x) represents the derivative of the numerator function u(x).

  True
• The Quotient Rule formula is dy/dx = (u'v - uv') / v²
• The numerator in the Quotient Rule is denoted as u(x)
• The Quotient Rule is used to find the derivative of a fractional function.
• Steps to find the derivative of y = (x² + 1) / (x - 2) using the Quotient Rule

  1️⃣ Identify u(x) and v(x)

  2️⃣ Calculate u'(x) and v'(x)

  3️⃣ Apply the Quotient Rule formula

  4️⃣ Simplify the expression
• What is the simplified derivative of (x² + 1) / (x - 2)?

  (x² - 4x - 1) / (x - 2)²
• What is the Quotient Rule formula for dy/dx if y = u(x) / v(x)?

  (u'v - uv') / v²
• What is the derivative of (x² + 1) / (x - 2) after simplification?

  (x² - 4x - 1) / (x - 2)²
• Match the function with its location in the Quotient Rule:

  Numerator ↔️ Above the division line

  Denominator ↔️ Below the division line
• Plugging the derivatives into the Quotient Rule formula is necessary to find dy/dx.


  True
• In the Quotient Rule, u(x) represents the numerator function.
• What does the Quotient Rule help us differentiate?

  Fractions of functions
• Steps to apply the Quotient Rule to y = (x² + 1) / (x - 2)

  1️⃣ Identify u(x) and v(x)

  2️⃣ Calculate u'(x) and v'(x)

  3️⃣ Apply the Quotient Rule formula

  4️⃣ Simplify the derivative expression
• What is the derivative of u(x) = x² + 1?

  2x
• Match the function with its derivative in the example y = (x² + 1) / (x - 2)

  u(x) = x² + 1 ↔️ u'(x) = 2x

  v(x) = x - 2 ↔️ v'(x) = 1
• What is the derivative of v(x) = x - 2?

  1
• The derivative of u(x) = x² + 1 is u'(x) = 2x.

  True
• The Quotient Rule applies when both the numerator and denominator are functions of x.
• Match the component of the Quotient Rule formula with its description:

  u(x) ↔️ Numerator function

  v(x) ↔️ Denominator function

  u'(x) ↔️ Derivative of u(x)

  v'(x) ↔️ Derivative of v(x)
• Steps to apply the Quotient Rule to differentiate y = (x² + 1) / (x - 2)

  1️⃣ Identify u(x) and v(x)

  2️⃣ Find u'(x) and v'(x)

  3️⃣ Substitute into the Quotient Rule formula

  4️⃣ Simplify the expression
• When using the Quotient Rule, the numerator is u(x).
• To use the Quotient Rule, you must first differentiate the numerator and denominator separately.
• After finding the derivatives, the next step is to plug them into the Quotient Rule formula.
• v'(x) is the derivative of the denominator function in the Quotient Rule.


  True
• u(x) in the Quotient Rule represents the denominator function.

  False
• The Quotient Rule formula can be written as **dy/dx = (u'v - uv') / v².
• In the example, after applying the Quotient Rule, dy/dx is equal to [(2x)(x - 2) - (x² + 1)(1)] / (x - 2)².
• The first step in applying the Quotient Rule is to identify the numerator and denominator functions.

  True
• For the function y = (x² + 1) / (x - 2), the numerator function u(x) is x² + 1
• What is the first step in simplifying a derivative expression after applying the Quotient Rule?

  Combine like terms
• Common mistakes when applying the Quotient Rule include errors in identifying the numerator or denominator
• The formula for the Quotient Rule is dy/dx = (u'v - uv') / v²
• What is the numerator function in the example y = (x² + 1) / (x - 2)?

  x² + 1
• The Quotient Rule can only be applied when both the numerator and denominator are functions of x.


  True
• Steps to apply the Quotient Rule to y = (x² + 1) / (x - 2)

  1️⃣ Identify u(x) and v(x)

  2️⃣ Calculate u'(x) and v'(x)

  3️⃣ Apply the Quotient Rule formula

  4️⃣ Simplify the derivative expression
• The Quotient Rule formula is dy/dx = (u'v - uv') / v²
• What is the derivative of (x² + 1) / (x - 2)?

  (x² - 4x - 1) / (x - 2)²
• What is the formula for the Quotient Rule?

  dy/dx = (u'v - uv') / v²
• If u(x) = x² + 1, then u'(x) = 2x.


  True
• The Quotient Rule is used to differentiate functions in the form of fractions.