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Applications of Derivatives Calculator

Monotonicity and extrema analyzer. Enter f(x) on [a,b] to scan the sign of f′ and get increasing/decreasing intervals and extrema—perfect for calculus monotonicity and extremum practice.

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f′ > 0 means increasing; f′ < 0 means decreasing; a sign change at f′ = 0 indicates an extremum
Supports + - * / ^ and sin cos tan sqrt exp log pi e
Start of interval
End of interval

📖 Tutorial｜Applications of Derivatives

1. Definition
f′>0 means increasing; f′<0 means decreasing. A point with f′=0 and a sign change is an extremum (left-negative/right-positive = min).
2. Symbols
SymbolMeaning
f′(x)The derivative
f′>0Increasing
f′<0Decreasing
f′=0Stationary point
3. How it works
  • We densely sample f′ on the interval;
  • Detect where f′ changes sign;
  • Report extrema and values.
4. Steps
  1. Enter f(x);
  2. Enter [a,b];
  3. Click Calculate.
5. Example
Example: f=x³−3x on [−3,3]. f′=3x²−3; at x=−1 sign changes +→− (max), at x=1 −→+ (min).
6. Pitfalls
f′=0 is not always an extremum (e.g. x³ at 0);
Check the sign actually changes;
Compare endpoints separately.

❓ FAQ｜Applications of Derivatives

Max or min?
f′ changes +→− means a max; −→+ means a min. Or check the second derivative.
Is f′=0 always an extremum?
No. f=x³ has f′=0 at x=0 but the sign does not change.
Are stationary points and extrema the same?
No. A stationary point has f′=0; an extremum also requires a sign change.
Do endpoints count?
Closed-interval endpoints are one-sided; compare them with interior extrema for the global max/min.
Why are the points approximate?
We scan a grid, so locations are accurate to the grid resolution.