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What is second derivative?

The second derivative is the derivative of the derivative — it measures how fast the gradient itself is changing, and hence the concavity of a curve.

Second derivative explained

Written d2ydx2\frac{d^{2}y}{dx^{2}} or f′′(x)f''(x), it tells you whether a curve bends upwards or downwards. In mechanics, differentiating displacement twice gives acceleration.
Its main exam use is classifying stationary points and identifying points of inflection where the second derivative changes sign.

Key formula

d2ydx2>0⇒\dfrac{d^{2}y}{dx^{2}} > 0 \Rightarrow minimum, d2ydx2<0⇒\dfrac{d^{2}y}{dx^{2}} < 0 \Rightarrow maximum

Worked example

If s=4t3s = 4t^{3} then velocity is 12t212t^{2} and acceleration is 24t24t, so at t=2t = 2 the acceleration is 48 m s−248\,\text{m s}^{-2}.

Examiner tip

A sign change in the second derivative — not just a zero value — is what confirms a point of inflection.

Second derivative: common questions

What does a positive second derivative mean?
The gradient is increasing, so the curve is concave up and any stationary point there is a minimum.
How is the second derivative used in mechanics?
Differentiating displacement once gives velocity and twice gives acceleration, which links directly to Newton's second law.

Related terms

Differentiation

Differentiation is the process of finding the rate at which one quantity changes with respect to another — the gradient of a curve at a point.

Stationary point

A stationary point is a point on a curve where the gradient is zero — a maximum, a minimum or a point of inflection.

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