Difference Quotient Calculator

How it is calculated

The tool works with the quadratic f(t) = a·t² + b·t + c. It evaluates f at your point x and at x + h for display, and the quotient is [f(x + h) − f(x)] ÷ h: the change in f divided by the change in the input, which is the slope of the line through the two points on the curve. For a quadratic this simplifies exactly to 2ax + b + ah, and the tool uses that form, so a very small h does not lose accuracy to subtracting two nearly equal numbers. All five values must be numbers and h may not be zero (a blank field or h = 0 shows no result). h can be negative, which looks back from x. Results show 4 decimals. As h shrinks the quotient tends to the derivative 2ax + b.

Common mistakes

• Setting h = 0. The difference quotient is a limit as h approaches 0, but at h = 0 the formula divides by zero and the tool shows no result. Use a small non-zero h such as 0.001.
• Reading f(x + h) in the output as proof the quotient is zero for tiny h. With h = 1e-12 the two function values both display as 25.0000, yet the quotient is still 13.0000 for the example, because it is computed from the simplified form.
• Treating the result as the derivative. It is the average slope between x and x + h, and equals 2ax + b only in the limit.
• Mixing up the roles of a, b and c: a belongs to x², b to x and c is the constant, which cancels out of the quotient entirely.
• Forgetting that a negative h looks backwards from x, to the point x − |h|, and gives a different quotient (for f(x) = x² at x = 3, h = −0.5 gives 5.5000 while h = 0.5 gives 6.5000).