Product Rule (DP IB Applications & Interpretation (AI): HL): Revision Note

Product rule

What is the product rule?

• The product rule states that if$y$ is the product of two functions of $x$ 

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

  • then
    

    $\frac{\mathbf{d}\mathit{y}}{\mathbf{d}\mathit{x}}\mathbf{=}\mathit{u}\frac{\mathbf{d}\mathit{v}}{\mathbf{d}\mathit{x}}\mathbf{+}\mathit{v}\frac{\mathbf{d}\mathit{u}}{\mathbf{d}\mathit{x}}$ 
    

    • This is given in the formula booklet

• In function notation this could be written as

$y=f\left(x\right)g\left(x\right)$

$\frac{\mathrm{d}y}{\mathrm{d}x}={f\left(x\right)g}^{\text{'}}\left(x\right)+{g\left(x\right)f}^{\text{'}}\left(x\right)$

• ‘Dash notation’ may be used as a shorter way of writing the rule

$y=uv$

$y\text{'}=uv\text{'}+vu\text{'}$

How do I know when to use the product rule?

• The product rule is used when we are trying to differentiate the product of two functions

  • these can easily be confused with composite functions (see chain rule)

    • $\sin \left(\cos x\right)$ is a composite function, “sin of cos of $x$”

    •  $\sin x\cos x$ is a product, “sin $x$ times cos $x$”

How do I use the product 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 product rule formula $\frac{\mathrm{d}y}{\mathrm{d}x}=u\frac{\mathrm{d}v}{\mathrm{d}x}+v\frac{\mathrm{d}u}{\mathrm{d}x}$
  Simplify the answer if straightforward to do so or if the question requires a particular form