f(x) = 2 + 5x − x^2

f(3 + h) − f(3) h

= 2 + 5(3 + h) − (3 + h)^2 − (2 + 15 − 9) h

= 2 + 5(3 + h) − (3 + h)^2 − 8 h

= 5h + 17 − (9 + 6h + h^2) − 8 h

= − h^2 − 9h + 8

= −(h^2 + 9h − 8)

f(x) = 2 + 5x - x²

f(3 + h) = 2 + 5(3 + h) - (3 + h)²

f(3 + h) = 2 + 15 + 5h - (9 + 6h + h²)

f(3 + h) = 17 + 5h - 9 - 6h - h²

f(3 + h) = -h² - h + 8

f(3) = 2 + 5(3) - (3)²

f(3) = 2 + 15 - 9

f(3) = 8

f(3 + h) - f(3) = -h² - h + 8 - 8

f(3 + h) - f(3) = -h² - h

f(3 + h) - f(3) = h(-h - 1)

Difference quotient:

[ f(3 + h) - f(3) ] / h = h(-h - 1) / h

= -h - 1, assuming h ≠ 0

The last step is to take the limit as h --> 0:

lim x->0 (-h - 1)

= 0 - 1

= -1

Let x = 3+h then

f(x) = 2 + 5*(3 + h) - (3+h)^2 = 17 + 5h - 9 - 6h - h^2 =8 - h - h^2

Now let x = 3 then

f(x) =2 +15 - 9 = 8 then

f(3+h) - f(3)*h = 8 - h - h^2 - 8h = 8 - 9h - h^2