The First Derivative & the Sign of f'(x)
Learn the first derivative and the sign of \(f'(x)\) for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-06). The sign of the derivative tells you which way a curve is heading: a positive value means the graph is rising and a negative value means it is falling.
You will learn to find where a function is increasing or decreasing by solving inequalities in \(f'(x)\), read the sign from a graph or table, and connect these intervals to the shape of the curve — the first step in sketching and in reading real-world trends.
Theory
The sign of \(f'(x)\) shows whether a function is increasing or decreasing. This Year 12 Mathematics Advanced topic (MAV-12-06) uses \(f'\) to describe behaviour.
\(f'(x)>0\): increasing; \(f'(x)<0\): decreasing; \(f'(x)=0\): a stationary point. To find where a function increases, solve the inequality \(f'(x)>0\).
Method
- Differentiate to get \(f'(x)\).
- Solve \(f'(x)>0\) (increasing) or \(<0\) (decreasing).
- Set \(f'(x)=0\) for stationary points.
| \(2x-4\) | \(>\) | \(0\) |
| \(x\) | \(>\) | \(2\) |
| \(2x-4\) | \(<\) | \(0\) |
| \(x\) | \(<\) | \(2\) |
| \(2x-4\) | \(=\) | \(0\) |
| \(x\) | \(=\) | \(2,\ f(2)=-4\) |
| \(f'(2)\) | \(=\) | \(3(4)=12>0\) |
Increasing.
Common pitfalls
Frequently asked questions
How do you tell if a function is increasing?
Its first derivative is positive there. Solve f prime of x greater than 0.
What does f prime equals zero mean?
The function is momentarily flat: a stationary point.
How do you find where a function is decreasing?
Solve the inequality f prime of x less than 0.
Does a bigger derivative mean increasing faster?
Yes, a larger positive derivative means a steeper upward slope, but any positive value means increasing.