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

1 Answer
Mar 7, 2017

Answer:

#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#