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Second Derivative

Second Derivative

The derivative of a derivative. Usually written f"(x), d squared y over dx squared — Leibniz notation for the second derivative, or y".

 

Example: f(x)=5x³−x²+13x−8; first derivative f′(x)=15x²−2x+13; second derivative f″(x)=30x−2

 

 

See also

Inflection point, second order critical point, second derivative test

Key Formula

f′′(x)=ddx ⁣[f′(x)]=d2ydx2f''(x) = \frac{d}{dx}\!\left[f'(x)\right] = \frac{d^2y}{dx^2}
Where:
  • f(x)f(x) = The original function
  • f′(x)f'(x) = The first derivative of f(x), representing the rate of change
  • f′′(x)f''(x) = The second derivative of f(x), representing the rate of change of the rate of change
  • d2ydx2\frac{d^2y}{dx^2} = Leibniz notation for the second derivative

Worked Example

Problem: Find the second derivative of f(x) = 3x⁴ − 5x² + 2x.
Step 1: Find the first derivative f'(x) using the power rule on each term.
f′(x)=12x3−10x+2f'(x) = 12x^3 - 10x + 2
Step 2: Differentiate f'(x) again to get the second derivative.
f′′(x)=36x2−10f''(x) = 36x^2 - 10
Step 3: Evaluate f''(x) at a specific point, say x = 1, to determine concavity there.
f′′(1)=36(1)2−10=26f''(1) = 36(1)^2 - 10 = 26
Step 4: Since f''(1) = 26 > 0, the graph of f is concave up at x = 1. The curve bends upward like a bowl.
Answer: The second derivative is f''(x) = 36x² − 10. At x = 1, f''(1) = 26 > 0, so the graph is concave up there.

Another Example

This example uses a trigonometric function instead of a polynomial and shows how the second derivative locates inflection points — a key application beyond simply computing the derivative.

Problem: Find the second derivative of g(x) = sin(x) and determine where inflection points occur on the interval [0, 2π].
Step 1: Differentiate g(x) = sin(x) to find the first derivative.
g′(x)=cos⁡(x)g'(x) = \cos(x)
Step 2: Differentiate again to find the second derivative.
g′′(x)=−sin⁡(x)g''(x) = -\sin(x)
Step 3: Set the second derivative equal to zero to find candidate inflection points.
−sin⁡(x)=0  ⟹  x=0,  π,  2π-\sin(x) = 0 \implies x = 0,\; \pi,\; 2\pi
Step 4: Check that the sign of g''(x) changes at each candidate. At x = π, g'' changes from negative (concave down on (0, π)) to positive (concave up on (π, 2π)), so x = π is an inflection point. At x = 0 and x = 2π, the sign also changes, so these are inflection points too.
Answer: g''(x) = −sin(x). Inflection points occur at x = 0, x = π, and x = 2π, where concavity switches.

Frequently Asked Questions

What does the second derivative tell you?
The second derivative tells you about the concavity of a function's graph. When f''(x) > 0, the graph is concave up (curves like a cup). When f''(x) < 0, the graph is concave down (curves like a cap). It also tells you whether the first derivative — the slope — is increasing or decreasing.
What is the difference between the first and second derivative?
The first derivative f'(x) gives the slope of the function, telling you how fast the output changes. The second derivative f''(x) gives the rate of change of that slope, telling you how the slope itself is changing. Think of position, velocity, and acceleration: the first derivative of position is velocity, and the second derivative of position is acceleration.
How do you use the second derivative to find inflection points?
Set f''(x) = 0 and solve for x. Then check whether f''(x) changes sign at each solution. If the sign switches from positive to negative or negative to positive, that x-value is an inflection point — a location where the graph changes from concave up to concave down, or vice versa. Simply having f''(x) = 0 is not enough; the sign must actually change.

First Derivative vs. Second Derivative

First DerivativeSecond Derivative
DefinitionRate of change of the original functionRate of change of the first derivative
Notationf'(x) or dy/dxf''(x) or d²y/dx²
Geometric meaningSlope of the tangent lineConcavity (curvature direction) of the graph
Physical exampleVelocity (rate of change of position)Acceleration (rate of change of velocity)
Key applicationFinding where a function increases or decreasesFinding inflection points; the second derivative test for extrema

Why It Matters

The second derivative appears constantly in calculus courses, especially in curve sketching, optimization, and the second derivative test for classifying local maxima and minima. In physics, the second derivative of position with respect to time is acceleration, making it fundamental to Newton's second law (F = ma). Engineers, economists, and scientists all rely on the second derivative to understand whether a quantity is speeding up, slowing down, or changing direction.

Common Mistakes

Mistake: Squaring the first derivative instead of differentiating it again.
Correction: The second derivative f''(x) means you apply the differentiation process to f'(x). It is NOT [f'(x)]². For example, if f'(x) = 3x², then f''(x) = 6x, not 9x⁴.
Mistake: Concluding there is an inflection point whenever f''(x) = 0.
Correction: f''(x) = 0 is a necessary condition for an inflection point, but not sufficient. You must verify that f''(x) changes sign at that point. For example, f(x) = x⁴ has f''(0) = 0, but f''(x) = 12x² ≥ 0 everywhere, so x = 0 is not an inflection point.

Related Terms