Question

DERIVATIVE HOMEWORK. ANSWER THE FOLLOWING QUESTION

Answer 1

By the product rule,

`(dy)/(dx) = (x-1)/(x+1) (d(7x^7-x^2))/(dx) + (7x^7-x^2)/(x+1) (d(x-1))/(x+1) + (7x^7-x^2)(x-1) \frac{d\left(\frac1{x+1}\right)}{dx}`

By the power and chain rules,

`(dy)/(dx) = ((x-1)(49x^6-2x))/(x+1) + (7x^7-x^2)/(x+1) - ((7x^7-x^2)(x-1))/((x+1)^2)`

When
`x=1`, the first and last term vanish, and we're left with

`(dy)/(dx)\bigg|_(x=1) = (7 - 1)/(1 + 1) = \frac62 = \boxed{3}`