Question

How do you differentiate `f(x)=(2+x )sqrt(2-3x` using the product rule?

Answer 1

`f'(x)=u'v+v'u`

Explanation:

Let `u=2+x` and `v=sqrt(2-3x)`
then `u'=1` and `v'=-3` x `1/2(2-3x)^(-1/2)`(chain rule)
Subbing into `f'(x)=u'v+v'u`,
`f'(x)=1(sqrt(2-3x))+(-3/2)(2-3x)^(-1/2)(2+x)`
Simplifying,
`f'(x)=sqrt(2-3x)-(3(2+x))/(2sqrt(2-3x))`