Question

F ( x ) = − x 2 − x . compute f ( a h ) − f ( a ) = assuming h ≠ 0 , compute f ( a h ) − f ( a ) h

Answer 1

Given

`f(x) = -x^2 - x`

we have

`f(a) = -a^2 - a`

`f(a+h) = -(a+h)^2 - (a+h) = -a^2 - 2ah - h^2 - a - h`

`\implies f(a+h) - f(a) = (-a^2 - 2ah - h^2 - a - h) - (-a^2 - a) = -2ah - h^2 - h`

`\implies \frac{f(a+h) - f(a)}h = \boxed{-2a - h - 1}`