Question

How do you differentiate `y(u)=(u^-2 + u^-3)(u^5 - 2u^2)` using the product rule?

Answer 1

`y'(u)=3u^2+2u-4u^-1-4u^-2+4u^-5+6u^-6`

Explanation:

differentiate using the `color(blue)"product rule"`

`"Given "y(u)=g(u).h(u)" then"`

`color(red)(bar(ul(|color(white)(2/2)color(black)(y'(u)=g(u)h'(u)+h(u)g'(u))color(white)(2/2)|)))larr" product rule"`

`"here "g(u)=u^-2+u^-3rArrg'(u)=-2u^-3-3u^-4`

`"and "h(u)=u^5-2u^-2rArrh'(u)=5u^4-4u`

`rArry'(u)=(u^-2+u^-3)(5u^4-4u)+(u^5-2u^-2)(-2u^-3-3u^-4)`

We can 'tidy up' by distributing the products.

`=5u^2-4u^-1+5u-4u^-2-2u^2-3u+4u^-5+6u^-6`

`=3u^2+2u-4u^-1-4u^-2+4u^-5+6u^-6`