Differentiate H(u) = (u - sqrt(u)) (u + sqrt(u))need an explanation

Question

can you explain each step like how you got (u^2-u)

Can you explain it by using the product rule .

Thanks !

Answer 1

I agree with Patrick B, but perhaps there is an easier way to see it

H(u)=(u-√u)(u+√u)=u² - u

(The sum of 2 numbers times the difference of the same 2 numbers = the difference of "their" squares.)

Then the derivative is obvious!

Answer 2

H = (u - u^(1/2)) ( u + u^(1/2))

dH/du = (u - u^(1/2))* (1 + (1/2)u^(-1/2)) + ( u + u^(1/2))(1 - (1/2)u^(-1/2)) <--- product rule

= u - u^(1/2) + (1/2)u^(-1/2)( u - u^(1/2) ) + (u + u^(1/2) - (1/2)u^(-1/2)(u + u^(1/2))

= u - u^(1/2) + (1/2)u^(1/2) - 1/2 + u + u^(1/2) - 1/2 u^(1/2)- 1/2

= 2u -1 ;

Now to check: if we FOIL it first: H = u^2 - u

so then dH/du = 2u - 1