The Difference Quotient

f(x+h) = 3(x+h)2 = 3(x2 + 2xh + h2) = 3x2 + 6xh + 3h2 You will start with a function f(x). For example, let f(x) = 3x2 First evaluate f(x+h) f(x+h) = 3(x+h)2 = 3(x2 + 2xh + h2) = 3x2 + 6xh + 3h2

Then subtract the original function from the expression you obtained Then subtract the original function from the expression you obtained . You will want to use parentheses because you may run into trouble without them. 3x2 + 6xh + 3h2 – (3x2)

Now simplify your new expression Now simplify your new expression. You will notice that something interesting happens. 3x2 + 6xh + 3h2 – (3x2) Simplifies to 6xh + 3h2

Now divide the expression by h. 6xh + 3h2 h

Simplify this expression by factoring an h from the numerator and cancelling it with the h in the denominator. 6xh + 3h2 = h h(6x + 3h)

Drum roll please Your final answer is 6x + 3h

Let’s try another. f(x) = x2 + 5x f(x+h) = (x+h)2 + 5(x+h) = x2 + 2xh + h2 + 5x + 5h Now subtract f(x) from the above expression x2 + 2xh + h2 + 5x + 5h – (x2 + 5x) = 2xh + h2 + 5h

Now divide by h 2xh + h2 + 5h = h h(2x + h + 5) = 2x + 5 + h