(i)
[tex]f(x) = x^2\\\\\frac{f(x +h) - f(x)}{h} = \frac{(x + h)^2 - x^2}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h} = \frac{h(2x + h)}{h} = 2x + h[/tex]
(ii)
[tex]f(x) = 3x + 5\\\\\frac{f(x + h) - f(x)}{h} = \frac{3(x + h) + 5 - (3x + 5)}{h} = \frac{3(x + h) + 5 - 3x - 5}{h} = \frac{3x + 3h + 5 - 3x - 5}{h} = \frac{3h}{h} = 3[/tex]
(iii)
[tex]f(x) = x^2 + 3x + 6\\\\\frac{f(x + h) - f(x)}{h} = \frac{(x + h)^2 + 3(x +h) + 6 - (x^2 + 3x + 6)}{h} = \frac{x^2 + 2xh + h^2 + 3x + 3h + 6 - x^2 - 3x - 6}{h} = \frac{2xh + h^2 + 3h}{h} = \frac{h(3x + h + 3)}{h} = 2x +h + 3[/tex]
(iv)
[tex]f(x) = x^3 + 1\\\\\frac{f(x + h) - f(x)}{h} = \frac{(x + h)^3 + 1 - (x^3 + 1)}{h} = \frac{(x + h)^3 + 1 - x^3 - 1}{h} = \frac{x^3 + 3x^2h + 3xh^2 + h^3 + 1 - x^3 - 1}{h} = \frac{3x^2h + 3xh^2 + h^3}{h} = \frac{h(3x^2 + 3xh + h^2)}{h} = 3x^2 + 3xh + h^2[/tex]
