Derivative definition
Define instantaneous rate of change as a limit of secant slopes.
Derivative definition is one of 2 limits formulas in the calculus section of this library, and it is used at ap · university level.
Why derivative definition works
A line through two points on the graph has slope equal to the rise over the run, and here the run is h. Shrinking h slides the second point toward the first, and those slopes settle on one number, which is the steepness of the curve right at x. The limit is what turns an average rate across an interval into a rate at a single instant.
What each symbol means
$h$ is an input change and $f'(x)$ is tangent slope.
Derivative definition: when it holds
The limit must exist and be finite for differentiability at $x$.
When it stops applying
At a corner the two sides disagree. For f(x) = |x| at x = 0 the quotient heads to 1 from the right and to −1 from the left, so no single limit exists and there is no derivative. A vertical tangent breaks it the same way, because the quotient grows past every bound.
Derivative definition: a worked example
For $f(x)=x^2$, the quotient becomes $2x+h$, so $f'(x)=2x$.
The mistake to avoid
What people do: Setting h to 0 straight away inside the difference quotient.
Why it goes wrong: That produces 0 divided by 0, which is not a number at all, so the quotient simply has no value at h equal to 0.
Do this instead: Simplify the fraction with algebra first so the h in the denominator cancels, and only then let h shrink toward 0.
Derivative definition: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $h$ is an input change and $f'(x)$ is tangent slope.
- Check the conditions before substituting. The limit must exist and be finite for differentiability at $x$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Limits
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where derivative definition comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Understanding Limits — the lesson behind this formula: describe the value a function approaches near a point.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about derivative definition
Why must h shrink toward 0 instead of just being tiny?
A fixed small h still reports an average slope across a short interval, and that answer keeps a leftover error term. Only the limit strips the error away and leaves the exact tangent slope.
What is the difference between this and the version that uses x − a in the denominator?
They describe the same tangent slope. One version names the gap between the two inputs, the other names the moving input itself, and both produce identical derivatives.
Can h be negative here?
Yes, and it must be allowed to be. Negative h puts the second point to the left of x, and the derivative exists only when the left and right answers agree.
Why does a constant function come out with derivative 0?
f(x+h) and f(x) are the same number, so the top of the fraction is 0 for every h. Zero divided by any nonzero h stays 0, and the limit of 0 is 0.