What the difference quotient is
The difference quotient is the slope of the secant line between x and x + h: the average rate of change of f over a step of size h. As h shrinks to 0, the secant turns into the tangent and the average rate becomes the instantaneous rate of change, the derivative f′(x).
f′(x) = limit as h → 0 of [f(x + h) − f(x)] / h
How it's simplified
For a polynomial or a ratio of polynomials, the calculator writes f(x + h) − f(x) over a common denominator, multiplies everything out, and cancels the factor h that every term of the top shares. What's left is valid for every h ≠ 0. Setting h = 0 in it gives f′(x). For other functions, like sin x or √x, there's no general algebraic simplification, so you get the numeric table instead.
Worked example
Take f(x) = x² + 3x.
- f(x + h) = (x + h)² + 3(x + h) = x² + 2xh + h² + 3x + 3h.
- f(x + h) − f(x) = 2xh + h² + 3h.
- Divide by h: the difference quotient is 2x + h + 3.
- Let h → 0: f′(x) = 2x + 3.
At x = 2 the quotient is 7.1 for h = 0.1 and 7.01 for h = 0.01, closing in on f′(2) = 7.
A rational example: for f(x) = 1/x, f(x + h) − f(x) = (x − (x + h))/(x(x + h)) = −h/(x(x + h)), so the quotient is −1/(x(x + h)) and f′(x) = −1/x².
Limitations
- Algebraic simplification covers polynomials and rational functions. Roots, trig, exponentials and logs get the numeric table only.
- In the table, very small h eventually loses accuracy to rounding: subtracting two nearly equal numbers wipes out digits. That's why the table stops at h = 0.000001.
- The simplified quotient is valid only for h ≠ 0, and only where f(x) and f(x + h) are both defined.