12.7PureCore

Increasing and Decreasing Functions

The sign of the derivative tells you whether a function is rising or falling. Positive gradient means increasing, negative means decreasing — so an inequality on dy/dx answers “where is the function increasing?”

25 min Video by Zeeshan Zamurred Differentiation

Edexcel AS Level Maths: 12.7 Increasing and Decreasing FunctionsWatch the full walkthrough before the notes below.

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What you'll be able to do

Link the sign of dy/dx to increasing/decreasing behaviour
Find intervals where a function is increasing
Find intervals where a function is decreasing
Prove a function is increasing for all x

The sign of the gradient

Where $\frac{d y}{d x} > 0$ the function is $increasing$ (sloping up); where $\frac{d y}{d x} < 0$ it is $decreasing$ (sloping down).

\frac{d y}{d x} > 0 \Rightarrow increasing, \frac{d y}{d x} < 0 \Rightarrow decreasing

Read the behaviour straight from the derivative’s sign.

Finding the intervals

Differentiate, then solve the inequality $\frac{d y}{d x} > 0$ (or $< 0$ ). The solution is the set of $x$ -values where the function increases (or decreases).

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

2Increasing where

2 x - 6 > 0

, i.e.

x > 3

Answerincreasing for

x > 3

Tip — It is a quadratic-inequality skill in disguise: differentiate, then solve the inequality.

Increasing for all x

To prove a function is increasing everywhere, show $\frac{d y}{d x} > 0$ for all $x$ — often by writing the derivative as a positive expression (e.g. a square plus a positive constant).

\frac{d y}{d x} = 3 x^{2} + 3

3 x^{2} \geq 0

, so

3 x^{2} + 3 \geq 3 > 0

always.

Answeralways increasing ∎

Formula recap

\frac{d y}{d x} > 0 \Rightarrow increasing

Positive gradient.

\frac{d y}{d x} < 0 \Rightarrow decreasing

Negative gradient.

\frac{d y}{d x} > 0 \forall x \Rightarrow always increasing

Increasing everywhere.

Common mistakes to avoid

Mixing up which sign means increasing.

Positive gradient = increasing; negative = decreasing.

Forgetting to differentiate before testing the sign.

It is the sign of dy/dx, not of y.

Key takeaways

dy/dx > 0 ⟶ increasing; dy/dx < 0 ⟶ decreasing.
Solve the inequality on dy/dx to find the intervals.
Show dy/dx > 0 for all x to prove a function always increases.

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