Quotient Rule (DP IB Analysis & Approaches (AA): HL): Revision Note

Quotient rule

What is the quotient rule?

• The quotient rule states that if $y$ is a quotient of functions of $x$

  • i.e. if $y=\frac{u}{v}$ where $u=u\left(x\right)$ and $v=v\left(x\right)$

  • then

    $\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}=\frac{\mathit{v}\frac{\mathbf{d}\mathit{u}}{\mathbf{d}\mathit{x}}-\mathit{u}\frac{\mathbf{d}\mathit{v}}{\mathbf{d}\mathit{x}}}{{\mathit{v}}^2}$ 
    

    • This is given in the formula booklet

• In function notation this could be written

$y=\frac{f\left(x\right)}{g\left(x\right)}$

$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{g\left(x\right)f^{\text{'}}\left(x\right)-f\left(x\right)g\text{'}\left(x\right)}{{\left[g\left(x\right)\right]}^2}$

• As with product rule, ‘dash notation’ may be used

$y=\frac{u}{v}$

$y\text{'}=\frac{vu\text{'}-uv\text{'}}{v^2}$

How do I know when to use the quotient rule?

• The quotient rule is used when trying to differentiate a fraction where both the numerator and denominator are functions of$\mathit{x}$

  • If the numerator is a constant, negative powers can be used

    • E.g. $\frac{2}{{\left(3x-7\right)}^2}=2{\left(3x-7\right)}^{-2}$

      • That can be differentiated using the chain rule

  • If the denominator is a constant, treat it as a factor of the expression

    • E.g. $\frac{{\left(3x-7\right)}^2}{2}=\frac{1}{2}{\left(3x-7\right)}^2$

      • That can be differentiated using the chain rule or by expanding the brackets

How do I use the quotient rule?

• STEP 1

  Identify the two functions,$u$ and$v$

  Differentiate both$u$ and$v$ with respect to$x$ to find$u\text{'}$ and$v\text{'}$
   

• STEP 2

  Obtain $\frac{\mathrm{d}y}{\mathrm{d}x}$ by applying the quotient rule formula$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{v\frac{\mathrm{d}u}{\mathrm{d}x}-u\frac{\mathrm{d}v}{\mathrm{d}x}}{v^2}$

  Simplify the answer if straightforward or if the question requires a particular form