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Differentiate a polynomial using the power rule.
Key values: f(x) = x^3 - 2x^2 + x · 1st derivative
Differentiate a composite function with sin(x^2).
Key values: f(x) = sin(x^2) · Chain rule applied
Differentiate exp(x)*cos(x) and evaluate the tangent at x = 0.
Key values: f(x) = exp(x)*cos(x) · Tangent at x = 0
Compute the second derivative to analyze concavity.
Key values: f(x) = x^4 - 6x^2 + 4 · 2nd derivative
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 |
The second derivative reveals the shape of a curve:
Second derivative test: At a critical point where : if , then is a local minimum. If , then is a local maximum.
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.
A particle's position: meters.
At : velocity = −1 m/s (moving backward), acceleration = 0 (inflection point in velocity), jerk = 6 (acceleration increasing steadily).
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.
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.
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.
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.
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.