Product Rule (Cambridge (CIE) O Level Additional Maths): Revision Note

Product rule

What is the product rule?

• The product rule is a formula that allows you to differentiate a product of two functions

• If $\mathit{y}\mathbf{=}\mathit{u}\mathbf{\times }\mathit{v}$ where u and v are functions of x then the product rule is:

$\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$

• In function notation, if $\mathbf{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{g}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{\times }\mathbf{h}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ then the product rule can be written as:

$\mathrm{f}\text{'}\left(x\right)=\mathrm{g}\left(x\right)\mathrm{h}\text{'}\left(x\right)+\mathrm{h}\left(x\right)\mathrm{g}\text{'}\left(x\right)$

• The easiest way to remember the product rule is, for $\mathit{y}\mathbf{=}\mathit{u}\mathbf{\times }\mathit{v}$ where u and v are functions of x:

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

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?

• Make it clear what$u,v,u\text{'}$ and$v\text{'}$ are

  • arranging them in a square can help

    • opposite diagonals match up

 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