AP Calculus AB and BC glossary

Difference quotient

Also called: Newton quotient

The difference quotient is the slope of the secant line joining two points on a curve, written as the change in output over the change in input. Taking its limit as the gap shrinks to zero produces the derivative.

f(x+h)−f(x)h\frac{f(x+h) - f(x)}{h}

The expression is exactly rise over run for the two points (x,f(x))(x, f(x)) and (x+h,f(x+h))(x+h, f(x+h)). Every piece of derivative notation traces back to it, which is why the AP exam still asks you to compute derivatives from the definition.

The alternate form f(x)−f(a)x−a\frac{f(x) - f(a)}{x - a} describes the same slope with the second point named directly rather than as an offset. Its limit as x→ax \to a gives f′(a)f'(a).

The mistake

Writing f(x+h)f(x+h) as f(x)+hf(x) + h. You must substitute x+hx+h into the whole rule, so for f(x)=x2f(x) = x^2 you get (x+h)2=x2+2xh+h2(x+h)^2 = x^2 + 2xh + h^2.

Appears in: Unit 2: Defining the Derivative