Question

How do you differentiate `g(x) = (x-1)(x-2)(x-3)` using the product rule?

Answer 1

The rule of product tells you that the derivative of a product of two functions is

`d/dx[f(x)g(x)]=(df(x))/dxg(x)+f(x)(dg(x))/dx`.

In our case the the product is triple, so we will first consider
`(x-1)*[(x-2)(x-3)]` as a product, then we will do the second part repeating the rule again.

`(dg(x))/dx=d/dx((x-1)[(x-2)(x-3)])`

`=(d(x-1))/dx[(x-2)(x-3)]+(x-1)(d[(x-2)(x-3)])/dx`

`=(x-2)(x-3)+(x-1)(d[(x-2)(x-3)])/dx`

now we reapply the same rule to the next product

`=(x-2)(x-3)+(x-1)((d(x-2))/dx(x-3)+(x-2)(d(x-3))/dx)`

`=(x-2)(x-3)+(x-1)((x-3)+(x-2))`

`=(x-2)(x-3)+(x-1)(x-3)+(x-1)(x-2)`.