12.8PureStretch

Second Order Derivatives

Differentiate the derivative and you get the second derivative — the rate of change of the gradient. It measures how a curve bends, and it is the key to classifying stationary points in the next lesson.

25 min Video by Zeeshan Zamurred Differentiation

Edexcel AS Level Maths: 12.8 Second Order DerivativesWatch the full walkthrough before the notes below.

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What you'll be able to do

Find the second derivative
Use the notation f″(x) and d²y/dx²
Interpret the sign of the second derivative
Connect it to concavity (bending) of a curve

Differentiating twice

The $second derivative$ is found by differentiating the first derivative again. It is written $\frac{d ^{2} y}{d x ^{2}}$ or $f^{''} (x)$ .

y = x^{4} \Rightarrow \frac{d y}{d x} = 4 x^{3} \Rightarrow \frac{d ^{2} y}{d x ^{2}} = 12 x^{2}

Apply the power rule a second time.

\frac{d y}{d x} = 6 x^{2} - 5

2Differentiate again:

12 x

Answer

\frac{d ^{2} y}{d x ^{2}} = 12 x

What it tells you

The second derivative is the rate of change of the gradient. A positive value means the gradient is increasing (the curve bends upwards, $\cup$ ); a negative value means it bends downwards, $\cap$ .

f^{''} (x) > 0 \Rightarrow \cup (concave up), f^{''} (x) < 0 \Rightarrow \cap (concave down)

Sign of the second derivative = the way the curve bends.

Tip — First derivative = slope; second derivative = how the slope itself is changing.

Formula recap

f^{''} (x) = \frac{d}{d x} (f^{'} (x))

Differentiate the derivative.

\frac{d ^{2} y}{d x ^{2}}

Leibniz notation.

f^{''} (x) > 0 \Rightarrow \cup, f^{''} (x) < 0 \Rightarrow \cap

Concavity.

Common mistakes to avoid

Writing the second derivative as (dy/dx)².

It is d²y/dx² — differentiate twice, not square the first derivative.

Stopping after one differentiation.

The second derivative needs the power rule applied a second time.

Key takeaways

The second derivative is the derivative of the derivative.
Notation: d²y/dx² or f″(x).
f″(x) > 0 ⟶ curve bends up (∪); f″(x) < 0 ⟶ bends down (∩).

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