Hello, Mr_Green!

Let f(x) = 5x³ - 6x² + 3

Simplify: .[f(x+h) - f(x-h)] / 2h

Don't be intimidated by a "difference quotient".

There are four separate steps to perform:

. . (1) Find f(x+h) . . . and simplify

. . (2) Find f(x-h) . . . and simplify

. . (3) Subtract (2) from (1)

. . (3) Divide by 2h . . . and simplify

(1) f(x+h) .= .5(x+h)³ - 6(x+h)² + 3

. . . . . . . . = .5(x³ + 3x²h + 3xh² + h³) - 6(x² + 2xh + h²) + 3

. . . . . . . . = .5x³ + 15x²h + 15xh² + 5h³ - 6x² - 12xh - 6h² + 3

(2) f(x-h) .= .5(h-h)3 - 6(x-h)2 + 3

. . . . . . . = .5(x³ - 3x²h + 3xh² - h³) - 6(x² - 2xh + h²) + 3

. . . . . . . =5x³ - 15x²h + 15xh² - 5h³ - 6x² + 12xh - 6h² + 3

(3) f(x+h) - f(x-h) .= .5x³ + 15x²h + 15xh² + 5h³ - 6x² - 12xh - 6h² + 3)

. . . . . . . . . . . . . . . - (5x³ -15x²h + 15xh² - 5h³ - 6x² + 12xh - 6h² + 3)

= .5x³ + 15x²h + 15xh² + 5h³ - 6x² - 12xh - 6h² + 3 - 5x³ + 15x²h - 15xh² + 5h³ + 6x² - 12xh + 6h² - 3

= .30x²h + 10h³ - 24xh

. . . f(x+h) - f(x) . . . .30x²h + 10h³ - 24xh

(4) --------------- . = . -------------------------

. . . . . . 2h . . . . . . . . . . . . .2h

. . . . . . . . . . . . . . .2h(15x² + 5h² - 12x)

Factor and reduce: . --------------------------

. . . . . . . . . . . . . . . . . . . . 2h

Answer: . 15x² + 5h² - 12x