2.9 The Quotient Rule

• What is the Quotient Rule used for?

  Differentiating quotients of functions
• The Quotient Rule formula is \frac{dy}{dx} = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^{2}}</latex>, where $y =$$\frac{u(x)}{v(x)}$ and $\frac{dy}{dx}$ is the derivative
• The Quotient Rule involves subtracting the product of the numerator's derivative and the denominator from the product of the numerator and the denominator's derivative.
• Steps to apply the Quotient Rule

  1️⃣ Identify $u(x)$ and $v(x)$

  2️⃣ Compute $u'(x)$ and $v'(x)$

  3️⃣ Substitute into the Quotient Rule formula

  4️⃣ Simplify the result
• What is the formula for the Quotient Rule?

  $\frac{dy}{dx} =$$\frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^{2}}$
• The Quotient Rule is used to differentiate functions that are the quotient of two other functions.
• What is the formula for the Sum/Difference Rule?

  $\frac{d}{dx}[u(x) \pm v(x)] =$$u'(x) \pm v'(x)$
• Match each differentiation rule with its formula:

  Sum/Difference Rule ↔️ $\frac{d}{dx}[u(x) \pm v(x)] =$$u'(x) \pm v'(x)$

  Product Rule ↔️ $\frac{d}{dx}[u(x)v(x)] =$$u'(x)v(x) +$$u(x)v'(x)$
• What is the formula for the Quotient Rule?

  \frac{dy}{dx} = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^{2}}</latex>
• The Quotient Rule is used to differentiate functions that are the quotient of two other functions.
• What is the formula for the Product Rule?

  $\frac{d}{dx}[u(x)v(x)] =$$u'(x)v(x) +$$u(x)v'(x)$
• The Quotient Rule formula includes subtracting the product of the numerator's derivative and the denominator from the product of the numerator and the denominator's derivative.
• What is the Quotient Rule formula?

  \frac{dy}{dx} = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^{2}}</latex>
• Match each differentiation rule with its formula:

  Sum/Difference Rule ↔️ $\frac{d}{dx}[u(x) \pm v(x)] =$$u'(x) \pm v'(x)$

  Product Rule ↔️ $\frac{d}{dx}[u(x)v(x)] =$$u'(x)v(x) +$$u(x)v'(x)$

  Quotient Rule ↔️ $\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] =$$\frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^{2}}$
• What is the general formula for the Quotient Rule when $y =$$\frac{u(x)}{v(x)}$?


  $\frac{dy}{dx} =$$\frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^{2}}$
• For the function y = \frac{x^{2}}{x + 1}</latex>, the derivative using the Quotient Rule is \frac{x^{2} + 2x}{(x + 1)^{2}}.
• What is the Quotient Rule used to differentiate?

  The quotient of two functions
• The Quotient Rule states that if $y =$$\frac{u(x)}{v(x)}$, then \frac{dy}{dx} = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^{2}}</latex>, where $\frac{dy}{dx}$ is the derivative
• For the function $y =$$\frac{x^{2}}{x + 1}$, the derivative using the Quotient Rule is \frac{x^{2} + 2x}{(x + 1)^{2}}.
• What type of functions is the Quotient Rule used to differentiate?

  Fractions of functions
• For y = \frac{x^{2}}{x + 1}</latex>, the derivative $\frac{dy}{dx}$ is \frac{x^{2} + 2x}{(x + 1)^{2}}
• The Quotient Rule formula is $\frac{dy}{dx} =$ \frac{u'(x)v(x) + u(x)v'(x)}{[v(x)]^{2}}.


  False
• What is the formula for differentiating a sum of functions using the Sum/Difference Rule?

  \frac{d}{dx}[u(x) \pm v(x)] = u'(x) \pm v'(x)</latex>
• The Product Rule states that $\frac{d}{dx}[u(x)v(x)] =$$u'(x)v(x) +$$u(x)v'(x)$. An example of its application is 2x \sin x + x^{2} \cos x
• Match the differentiation rule with its formula:

  Sum/Difference Rule ↔️ $\frac{d}{dx}[u(x) \pm v(x)] =$$u'(x) \pm v'(x)$

  Product Rule ↔️ $\frac{d}{dx}[u(x)v(x)] =$$u'(x)v(x) +$$u(x)v'(x)$

  Quotient Rule ↔️ $\frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^{2}}$
• What is an example of a derivative calculated using the Quotient Rule?

  \frac{2x(x + 1) - x^{2}}{(x + 1)^{2}}</latex>
• For $y =$$\frac{x^{2}}{x + 1}$, the derivative $\frac{dy}{dx}$ using the Quotient Rule is \frac{x^{2} + 2x}{(x + 1)^{2}}
• The Quotient Rule is used to differentiate functions of the form y = \frac{u(x)}{v(x)}
• What is the formula for the Quotient Rule?

  \frac{dy}{dx} = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^{2}}</latex>
• What is the derivative of $2x +$$3$ using the Sum/Difference Rule?


  $2$
• The Product Rule states that $\frac{d}{dx}[u(x)v(x)] =$$u'(x)v(x) +$$u(x)v'(x)$, where $u'(x)$ is the derivative of u(x)
• Give an example where the Quotient Rule is used to find the derivative of \frac{x^{2}}{x + 1}</latex>

  \frac{x^{2} + 2x}{(x + 1)^{2}}
• The Quotient Rule is used to differentiate functions that are the sum of two other functions.

  False
• The Quotient Rule is used to differentiate functions that are the quotient of two other functions.
• What does $u'(x)$ represent in the Quotient Rule?


  Derivative of the numerator
• Steps to apply the Quotient Rule:

  1️⃣ Identify $u(x)$ and $v(x)$.

  2️⃣ Compute $u'(x)$ and $v'(x)$.

  3️⃣ Use the formula: $\frac{d}{dx}\left(\frac{u}{v}\right) =$$\frac{u'v - uv'}{v^{2}}$.

  4️⃣ Simplify the resulting expression.
• The derivative of $v(x) =$$x +$$1$ is $v'(x) =$$2$.


  False
• The Quotient Rule states that if $y =$$\frac{u(x)}{v(x)}$, then \frac{dy}{dx} = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^{2}}</latex>, where $v'(x)$ is the derivative of the denominator
• What is the derivative of the denominator in the Quotient Rule called?

  $v'(x)$
• To apply the Quotient Rule, we use the formula \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^{2}}