What does the second derivative tell you?

How the slope is changing. A positive second derivative means the slope is increasing, so the curve bends upwards; a negative one means it bends downwards.

Are third derivatives in DSE M1?

No. Third and higher order derivatives, and the second derivative of an implicit function, are not required.

How do I tell concave upwards from concave downwards on a graph?

Draw a tangent. If the curve lies above its tangent it is concave upwards; if it lies below, it is concave downwards.

The Second Derivative and Concavity (DSE M1)

The second derivative f″(x) (also y″ or d²y/dx²) is the derivative of f′(x). Where f″(x) > 0 the curve is concave upwards; where f″(x) < 0 it is concave downwards.

Solve Second Derivative and Concavity questions with AI

f(x) = x³ − 3x² from Example 1. The tangent lies above the curve while x < 1 (concave downwards) and below it once x > 1 (concave upwards). Try it yourself →

Second Derivative: Key Facts

Notationf″(x), y″, d²y/dx²

Concave upwardsf″(x) > 0

Concave downwardsf″(x) < 0

Second derivative testAt f′ = 0: f″ > 0 minimum, f″ < 0 maximum

Worked Examples — Second Derivative and Concavity

Example 1: Concavity of a cubic

Let f(x) = x³ − 3x². Find f″(x) and determine where the graph is concave upwards and where it is concave downwards.

f′(x) = 3x² − 6x
f″(x) = 6x − 6
f″(x) > 0 when x > 1: concave upwards
f″(x) < 0 when x < 1: concave downwards

Answer: f″(x) = 6x − 6; concave downwards for x < 1, concave upwards for x > 1

Example 2: Second derivative with an exponential

Find d²y/dx² if y = x e⁻ˣ.

dy/dx = e⁻ˣ − x e⁻ˣ = (1 − x)e⁻ˣ
d²y/dx² = −e⁻ˣ − (1 − x)e⁻ˣ
= (x − 2)e⁻ˣ

Answer: d²y/dx² = (x − 2)e⁻ˣ

💡 Exam Traps & Secrets

⚠️ d²y/dx² is not (dy/dx)²The second derivative means "differentiate again", not "square the first derivative". For y = x³, d²y/dx² = 6x but (dy/dx)² = 9x⁴.

⚠️ Concavity decides the trapezoidal ruleThe sign of f″ is how you justify whether a trapezoidal estimate is an over-estimate (concave upwards) or an under-estimate (concave downwards).

✏️ Self-Test: Quick Interactive Practice

How to study: Click any question below to instantly load it on the coordinate grid and check your steps.

(Concavity)Find the second derivative of y = x⁴ − 6x² and the intervals where the curve is concave upwards.

(Second derivative)Find d²y/dx² for y = ln(x² + 1).

(Sign of f″)Show that y = e^(2x) is concave upwards for all x.