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Difference Quotient Calculator

Simplify the difference quotient [f(x + h) − f(x)] / h for any polynomial step by step: f(x + h) expanded, the simplified quotient, its limit f′(x) as h → 0 and a numeric value.

f(x)

Value of x (numeric check)

Value of h (numeric check)

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How it works

Difference quotient = [f(x + h) − f(x)] / h · expand f(x + h), subtract f(x), divide every remaining term by h · lim (h → 0) = f′(x)

The difference quotient is the slope of the secant line between the points (x, f(x)) and (x + h, f(x + h)). For a polynomial every term of f(x + h) − f(x) contains a factor h, so it cancels exactly and leaves a polynomial in x and h. Letting h shrink to 0 gives the derivative, which is why the difference quotient is the first step of the limit definition of the derivative in calculus (OpenStax Calculus Vol. 1 §3.1). The expansion here is exact, with fractions kept as fractions.

Worked example

f(x) = x² + 3x − 1. f(x + h) = (x + h)² + 3(x + h) − 1 = x² + 2xh + h² + 3x + 3h − 1. Subtracting f(x) leaves 2xh + h² + 3h, and dividing by h gives 2x + h + 3. As h → 0 this becomes f′(x) = 2x + 3. Numeric check at x = 1, h = 0.5: [f(1.5) − f(1)] / 0.5 = (5.75 − 3) / 0.5 = 5.5.

Sources: OpenStax Calculus Vol. 1 — 3.1 Defining the Derivative · Paul’s Online Math Notes — The Definition of the Derivative

Frequently asked questions

What is the difference quotient?

It is [f(x + h) − f(x)] / h, the average rate of change of f over the small interval from x to x + h, i.e. the slope of a secant line.

How do you simplify a difference quotient?

Expand f(x + h), subtract f(x) so the terms without h cancel, factor h out of what remains and cancel it with the denominator. Never cancel before subtracting.

Why can h not be 0?

The expression divides by h. After simplifying you can let h approach 0, which gives the derivative f′(x), but you cannot plug h = 0 into the original fraction.

What is the difference quotient of x²?

[(x + h)² − x²] / h = (2xh + h²) / h = 2x + h, whose limit as h → 0 is 2x.

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This calculator is for educational purposes. Double-check important results before acting on them.