Sunday, 24 November 2013

Generalised Product Rule

When differentiating some products, the two and three unit courses generally use binomials in the product.  Occasionally, a product will arise that has three components; how can this be done?

Fortunately, there is an easy way to do this:
Consider $y=uvw$, where $u=u(x), v=v(x)$ and $w=w(x)$.  The derivative is then
$\frac{\mathrm{d}}{\mathrm{d}x}(uvw)=\frac{\mathrm{d}u}{\mathrm{d}x}vw+u\frac{\mathrm{d}v}{\mathrm{d}x}w+uv\frac{\mathrm{d}w}{\mathrm{d}x}$

The product can also be extended to any number of terms in the product.  It looks scary, but never let notation scare you in maths!

It should also be noted that doesn't appear in the syllabus documents - it just may make your life that little bit easier!