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Higher Order Derivative Calculator
Find the 2nd, 3rd, 4th, or 5th derivative of any function. Supports second derivative f''(x) for concavity analysis and Taylor coefficients.
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Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Power Rule
Differentiate a polynomial using the power rule.
Key values: f(x) = x^3 - 2x^2 + x · 1st derivative
Chain Rule
Differentiate a composite function with sin(x^2).
Key values: f(x) = sin(x^2) · Chain rule applied
Product Rule with Tangent
Differentiate exp(x)*cos(x) and evaluate the tangent at x = 0.
Key values: f(x) = exp(x)*cos(x) · Tangent at x = 0
Second Derivative
Compute the second derivative to analyze concavity.
Key values: f(x) = x^4 - 6x^2 + 4 · 2nd derivative
What Are Higher-Order Derivatives?
The second derivative is the derivative of the first derivative. It measures how the rate of change itself is changing. Each successive derivative peels back another layer of the function's behavior.
| Derivative | Notation | Measures | Physics Analogy |
|---|---|---|---|
| 1st | Slope / rate of change | Velocity | |
| 2nd | Concavity / curvature | Acceleration | |
| 3rd | Rate of curvature change | Jerk | |
| 4th | Smoothness measure | Snap |
Concavity and Inflection Points
The second derivative reveals the shape of a curve:
- : concave up (cup shape) — the function curves upward
- : concave down (cap shape) — the function curves downward
- (with sign change): inflection point — concavity switches
Second derivative test: At a critical point where : if , then is a local minimum. If , then is a local maximum.
Taylor Series Connection
Higher-order derivatives are the building blocks of Taylor series, which approximate functions as infinite polynomials:
The more derivatives you include, the better the polynomial approximates the original function near . This calculator computes up to the 5th derivative, giving you the first six terms of the Taylor expansion.
Worked Examples
Example: Motion Analysis
A particle's position: meters.
- Velocity: m/s
- Acceleration: m/s²
- Jerk: m/s³ (constant)
At : velocity = −1 m/s (moving backward), acceleration = 0 (inflection point in velocity), jerk = 6 (acceleration increasing steadily).
Frequently Asked Questions
What is a second derivative?
The second derivative f''(x) is the derivative of the first derivative. It measures how the rate of change itself is changing. In physics, if position gives the first derivative as velocity, the second derivative is acceleration.
How does the second derivative determine concavity?
When f''(x) > 0, the function is concave up (curves upward like a cup). When f''(x) < 0, it is concave down (curves downward like a cap). Where f''(x) = 0 with a sign change, there is an inflection point where concavity switches.
What is the second derivative test for maxima and minima?
At a critical point where f'(c) = 0: if f''(c) > 0, then c is a local minimum. If f''(c) < 0, then c is a local maximum. If f''(c) = 0, the test is inconclusive and you must use the first derivative test instead.
How do higher-order derivatives connect to Taylor series?
Higher-order derivatives are the coefficients of Taylor series: f(x) = sum of f^(n)(a)/n! times (x-a)^n. The more derivatives you include, the better the polynomial approximation near x = a.
What do the third and fourth derivatives measure in physics?
The third derivative of position is called jerk (the rate of change of acceleration). The fourth derivative is called snap. These are important in mechanical engineering and ride comfort analysis for vehicles and roller coasters.
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