What is the second derivative?
The second derivative is obtained by differentiating the first derivative again. It is written d2y/dx2 or f″(x). If the first derivative gives the gradient of a curve, the second derivative describes how that gradient is changing.
Key formula
d2y/dx2 = d/dx(dy/dx) · Second-derivative test at a stationary point: if d2y/dx2 > 0 → minimum, if d2y/dx2 < 0 → maximum
Testing turning points
At a turning point the first derivative is zero (dy/dx = 0). To decide its nature, substitute the x-value into the second derivative. A positive result means the curve is concave up, giving a minimum; a negative result means concave down, giving a maximum. If the second derivative is zero, the test is inconclusive and other methods are needed.
Worked example
Let y = x3 − 3x2. Then dy/dx = 3x2 − 6x and d2y/dx2 = 6x − 6. At x = 2, d2y/dx2 = 12 − 6 = 6, which is positive, so x = 2 gives a minimum. At x = 0, d2y/dx2 = −6, which is negative, so x = 0 gives a maximum.
Repeated differentiation
Differentiating term by term is the quickest method. For y = x4, dy/dx = 4x3 and d2y/dx2 = 12x2.
Remember
- Differentiate twice to get d2y/dx2.
- d2y/dx2 > 0 means minimum; < 0 means maximum.
- The test needs the stationary point first (dy/dx = 0).