Second Derivatives

This lesson covers the second derivative d²y/dx², its use in determining the nature of stationary points, concavity, and points of inflection, as required by the Edexcel A-Level Mathematics specification (9MA0).

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What Is the Second Derivative?

The second derivative is the derivative of the derivative:

d²y/dx² = d/dx(dy/dx)

If y = f(x), then:

• f'(x) = dy/dx = the first derivative (gradient / rate of change)
• f''(x) = d²y/dx² = the second derivative (rate of change of the gradient)

Worked Example 1

Find d²y/dx² for y = 3x⁴ - 2x³ + 5x.

Solution: dy/dx = 12x³ - 6x² + 5

d²y/dx² = 36x² - 12x = 12x(3x - 1)

Worked Example 2

Find f''(x) for f(x) = eˣ + ln x.

Solution: f'(x) = eˣ + 1/x f''(x) = eˣ - 1/x² = eˣ - x⁻²

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Determining the Nature of Stationary Points

A stationary point occurs where dy/dx = 0. The second derivative test tells you the nature: