The First Derivative & the Sign of f'(x)
The first derivative shows where a function increases or decreases: \(f'(x)>0\) means increasing and \(f'(x)<0\) means decreasing.
Part of the NSW Year 12 Mathematics Advanced course, in the Calculus area of study (Applications of calculus focus area) of the 2024 syllabus. Work through practice questions with fully worked solutions and video lessons, or scroll down for the theory summary and worked examples.
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.