| Input | Result | |
|---|---|---|
| f(x)=x⁴−3x² | f''(x)=12x²−6 | |
| f(x)=sin(x) | f''(x)=−sin(x) | |
| f(x)=eˣ | f''(x)=eˣ | |
| f(x)=ln(x) | f''(x)=−1/x² |
The Second Derivative — What It Tells You
The second derivative f''(x) is the derivative of f'(x). It measures the rate of change of the slope — geometrically, this is concavity. f''(x) > 0: the curve is concave up (bowl shape, slope increasing). f''(x) < 0: the curve is concave down (hill shape, slope decreasing). f''(x) = 0: possible inflection point — check if sign of f'' actually changes.
Physical interpretation: If s(t) = position, then s'(t) = velocity and s''(t) = acceleration. The second derivative converts position into acceleration — Newton's F=ma is m·s''(t) = F(t).
Second Derivative Test for Extrema: At a critical point c where f'(c) = 0: f''(c) > 0 → local minimum (concave up, like the bottom of a bowl). f''(c) < 0 → local maximum (concave down, like the top of a hill). f''(c) = 0 → inconclusive, use the first derivative test.
Examples: f(x)=x⁴−3x²: f'(x)=4x³−6x, f''(x)=12x²−6. At x=0: f''(0)=−6<0 (local max). At x=±√(1/2): f''=0 (inflection points). f(x)=sin(x): f''(x)=−sin(x). f''(x)=0 at x=nπ — inflection points where concavity switches.
How to Use This Second Derivative Calculator
Enter your expression in the input box above using standard mathematical notation. Use ^ for exponents (e.g., x^3 for x³), * for multiplication when needed, sin(), cos(), tan(), ln(), sqrt() for standard functions. Then click Calculate to get your answer with full step-by-step working.
This calculator handles polynomial, trigonometric, exponential, logarithmic expressions, and combinations thereof. Results are shown in simplified form where possible, with each step of the working displayed below the answer.
For best results, enter expressions clearly without ambiguity. Use parentheses to group terms: (x^2 + 1)/(x - 1) rather than x^2+1/x-1. The calculator follows standard order of operations.
What the Second Derivative Really Measures
The first derivative f'(x) measures the rate of change of f — how fast the output changes as the input changes. The second derivative f''(x) measures the rate of change of that rate — how fast the slope is changing. This sounds abstract, but the physical interpretation is concrete: if s(t) is position, s'(t) is velocity (rate of position change), and s''(t) is acceleration (rate of velocity change). The second derivative is what you feel when a car accelerates or brakes — it's the change in speed, not the speed itself.
Geometrically, f''(x) measures concavity. Where f'' > 0, the slope is increasing — the curve bends upward like a bowl. Where f'' < 0, the slope is decreasing — the curve bends downward like a hill. This curvature information is what transforms a set of critical points into a complete picture of a function's shape. Knowing where f' = 0 tells you where the function is flat; knowing f'' at those points tells you whether they are peaks, valleys, or saddle points.
The Second Derivative Test — Proof Sketch
Why does f''(c) > 0 at a critical point imply a local minimum? At a critical point, f'(c) = 0 (slope is zero). f''(c) > 0 means f' is increasing at c — so just to the left of c, f' < 0 (slope was negative, function was falling), and just to the right, f' > 0 (slope becomes positive, function is rising). A function that falls then rises has a valley — a local minimum. This is exactly the first derivative test conclusion, derived from the second derivative's sign.
Similarly, f''(c) < 0 means f' is decreasing at c — slope goes from positive (rising) to negative (falling), creating a peak — a local maximum. When f''(c) = 0, we cannot determine the sign behavior of f' near c from this information alone — the test is genuinely inconclusive, not just "insufficient data."
Concavity in Applications
Concavity has direct meaning in applied contexts. In economics, the second derivative of a production function f''(K) < 0 means diminishing marginal returns — each additional unit of capital adds less output than the previous one. This is an empirical regularity built into most economic models. In machine learning, the Hessian matrix (matrix of all second partial derivatives of the loss function) determines whether a critical point is a local minimum of the loss — a positive definite Hessian means it is. Second-order optimisation methods (Newton's method) use the Hessian to make faster steps toward the minimum than gradient descent, at the cost of computing all these second partial derivatives.
- Stewart, J. (2015). Calculus, §4.3. Cengage.
- Apostol, T. (1967). Calculus, Vol. 1, Ch. 4. Wiley.
- Larson, R. & Edwards, B. (2013). Calculus, 10th ed. Cengage.