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Definition of Derivative Calculator

Derivative-definition demo. Enter f(x) and x0 to watch the difference quotient [f(x0+h)−f(x0)]/h shrink toward the tangent slope—for intuitive understanding.

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f′(x0) = lim(h→0) [f(x0+h)−f(x0)] / h
Supports + - * / ^ and sin cos tan sqrt exp log pi e
Compute the derivative here

📖 Tutorial｜Definition of Derivative

1. Definition
The derivative f′(x0) = lim(h→0)[f(x0+h)−f(x0)]/h, geometrically the slope of the tangent line.
2. Symbols
SymbolMeaning
f(x)The function
x0Point of differentiation
hIncrement
f′(x0)Derivative value (tangent slope)
3. How it works
  • We compute the quotient at h=0.1,0.01,0.001;
  • Watch it settle to a value as h shrinks;
  • That value approximates the derivative.
4. Steps
  1. Enter f(x), e.g. x^2;
  2. Enter x0, e.g. 3;
  3. Click Calculate to watch the quotient approach f′(x0).
5. Example
Example: f(x)=x² at x0=3. The quotient [(3+h)²−9]/h=6+h→6, so f′(3)=6.
6. Pitfalls
h cannot be 0 but should be small enough to approximate the limit;
Left and right difference quotients must agree;
Very small h causes loss of significance.

❓ FAQ｜Definition of Derivative

What is the geometric meaning?
f′(x0) is the slope of the tangent at (x0,f(x0)), the instantaneous rate of change.
Why shrink h?
The derivative is a limit; only as h→0 does the quotient equal the tangent slope.
When does the derivative not exist?
Corners, cusps, vertical tangents, mismatched one-sided derivatives, discontinuity.
Why the small gap from the formula?
We use a finite h, so there is truncation error; at h=0.001 it is very close.
How does the quotient relate to derivative rules?
The quotient is the definition; the power/chain rules are its simplifications.