2.9 The Quotient Rule

• The Quotient Rule states that the derivative of a fraction is equal to the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.


  True
• What is the formula for the Power Rule?

  $\frac{d}{dx}x^{n} =$$nx^{n - 1}$
• What is the formula for the Quotient Rule?

  \frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{g(x)\frac{d}{dx}f(x) - f(x)\frac{d}{dx}g(x)}{[g(x)]^{2}}</latex>
• What is the derivative of $f(x)g(x)$ using the Product Rule?


  $f'(x)g(x) +$$f(x)g'(x)$
• What is the formula for the Quotient Rule in its general form?

  $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) =$$\frac{g(x)\frac{d}{dx}f(x) - f(x)\frac{d}{dx}g(x)}{[g(x)]^{2}}$
• The Quotient Rule states that the derivative of a fraction is equal to the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator
• Steps to apply the Quotient Rule

  1️⃣ Identify the numerator $f(x)$ and denominator $g(x)$

  2️⃣ Find the derivatives $f'(x)$ and $g'(x)$

  3️⃣ Substitute into the Quotient Rule formula

  4️⃣ Simplify the result if necessary
• In a fraction, the numerator appears above the division line, while the denominator appears below
• Match the differentiation rule with its formula:

  Power Rule ↔️ $\frac{d}{dx}x^{n} =$$nx^{n - 1}$

  Product Rule ↔️ $\frac{d}{dx}(f(x)g(x)) =$$f'(x)g(x) +$$f(x)g'(x)$

  Quotient Rule ↔️ $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) =$$\frac{g(x)\frac{d}{dx}f(x) - f(x)\frac{d}{dx}g(x)}{[g(x)]^{2}}$
• The Power Rule formula is \frac{d}{dx}x^{n} = nx^{n - 1}</latex>

  True
• Steps to differentiate a fraction using the Quotient Rule

  1️⃣ Identify the numerator $f(x)$ and denominator $g(x)$

  2️⃣ Apply the Quotient Rule formula
• In the example y = \frac{x^{2}}{x + 1}</latex>, what are the numerator and denominator functions?

  $f(x) =$$x^{2}$ and $g(x) =$$x +$$1$
• In the Quotient Rule formula, what does $f(x)$ represent?


  Numerator
• Match the differentiation rule with its formula and usage:

  Power Rule ↔️ $\frac{d}{dx}x^{n} =$$nx^{n - 1}$ ||| Differentiating powers like $x^{3}$

  Product Rule ↔️ $\frac{d}{dx}(f(x)g(x)) =$$f'(x)g(x) +$$f(x)g'(x)$ ||| Differentiating the product of two functions

  Quotient Rule ↔️ $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) =$$\frac{g(x)\frac{d}{dx}f(x) - f(x)\frac{d}{dx}g(x)}{[g(x)]^{2}}$ ||| Differentiating fractions like $\frac{x^{2}}{x + 1}$
• What is the Quotient Rule used for?

  Differentiating a fraction
• The numerator in the Quotient Rule formula is denoted by f(x)
• What is the denominator in the Quotient Rule formula squared?

  [g(x)]^{2}</latex>
• What is the formula for the Power Rule?

  $\frac{d}{dx}x^{n} =$$nx^{n - 1}$
• Match the differentiation rule with its usage:

  Power Rule ↔️ Differentiating powers like $x^{3}$

  Product Rule ↔️ Differentiating products of two functions

  Quotient Rule ↔️ Differentiating fractions
• What is the derivative of $y =$$\frac{x^{2}}{x + 1}$ using the Quotient Rule?


  \frac{x^{2} + 2x}{(x + 1)^{2}}
• In the function y = \frac{x^{2}}{x + 1}</latex>, what is the numerator $f(x)$?


  $x^{2}$
• In the function $y =$$\frac{x^{2}}{x + 1}$, $f(x) =$$x^{2}$ and $g(x) =$$x +$$1$True
• Steps to differentiate a fraction using the Quotient Rule

  1️⃣ Identify the numerator $f(x)$ and denominator g(x)</latex>

  2️⃣ Apply the Quotient Rule formula

  3️⃣ Simplify the resulting expression
• When differentiating $y =$$\frac{x^{2}}{x + 1}$, $f(x) =$$x^{2}$ and $g(x) =$$x +$$1$


  True
• What formula is used to differentiate a fraction in calculus?

  Quotient Rule
• Steps to apply the Quotient Rule

  1️⃣ Identify the numerator $f(x)$

  2️⃣ Identify the denominator $g(x)$

  3️⃣ Apply the Quotient Rule formula
• After applying the Quotient Rule to y = \frac{x^{2}}{x + 1}</latex>, the derivative is \frac{x^{2} + 2x}{(x + 1)^{2}}
• Match the simplification method with its example:

  Factoring ↔️ \frac{2x^{2} + 2x}{(x + 1)^{2}} =$\frac{2x(x + 1)}{(x + 1)^{2}}$

  Combining Like Terms ↔️ $2x^{2} +$$3x^{2} =$$5x^{2}$

  Reducing Fraction ↔️ $\frac{2x(x + 1)}{(x + 1)^{2}} =$$\frac{2x}{x + 1}$
• The derivative of $y =$$\frac{\sin x}{x}$ is \frac{x \cos x - \sin x}{x^{2}}
• What is the Quotient Rule used for in calculus?

  Differentiating fractions
• The Quotient Rule is contrasted with the Power Rule and the Product Rule.
• What is the formula for the Product Rule?

  $\frac{d}{dx}(f(x)g(x)) =$$f'(x)g(x) +$$f(x)g'(x)$
• The derivative of a fraction is equal to the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
• What is the derivative of $x^{n}$ using the Power Rule?


  $nx^{n - 1}$
• The Quotient Rule is applicable when the denominator of a fraction is a constant.

  False
• What is the formula for the Quotient Rule?

  $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) =$$\frac{g(x)\frac{d}{dx}f(x) - f(x)\frac{d}{dx}g(x)}{[g(x)]^{2}}$
• The Quotient Rule is used to differentiate products of functions.

  False
• What is the derivative of $y =$$\frac{x^{2}}{x + 1}$ using the Quotient Rule?


  $y' =$$\frac{(x + 1)(2x) - x^{2}(1)}{(x + 1)^{2}}$
• What are the numerator and denominator functions in $y =$$\frac{x^{2}}{x + 1}$?


  f(x) = x^{2}, g(x) = x + 1</latex>
• The Quotient Rule states that the derivative of a fraction is equal to the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator