The Second Derivative
Learn the second derivative for NSW Year 12 Mathematics Advanced (NESA outcome MAV-12-06). Differentiating twice measures how quickly the gradient itself is changing, telling you whether a curve bends upwards or downwards and, for motion, giving acceleration.
You will learn to find second derivatives using \(f''(x)\) and other notation, interpret the sign as concave up or concave down, and use it to classify stationary points — knowledge that drives curve sketching, optimisation and motion problems.
Theory
The second derivative is the derivative of the derivative, measuring concavity. This Year 12 Mathematics Advanced topic (MAV-12-06) finds and interprets \(f''(x)\).
\(f''(x)=\dfrac{d^{2}y}{dx^{2}}\). \(f''>0\) is concave up (holds water); \(f''<0\) is concave down. In motion, \(f''\) is acceleration.
Method
- Differentiate twice.
- Read the sign of \(f''\) for concavity.
- In motion, \(f''\) is the acceleration.
| \(f'(x)\) | \(=\) | \(3x^{2}\) |
| \(f''(x)\) | \(=\) | \(6x\) |
| \(f''(2)\) | \(=\) | \(12\) |
\(f''(2)=12>0\), so concave up.
| \(v\) | \(=\) | \(3t^{2}-12t\) |
| \(a\) | \(=\) | \(6t-12\) |
Common pitfalls
Frequently asked questions
What is the second derivative?
It is the derivative of the first derivative, written f double prime or d squared y by dx squared.
What does the second derivative tell you?
It gives the concavity: positive means concave up, negative means concave down. In motion it is the acceleration.
How do you find the second derivative?
Differentiate the function once to get f prime, then differentiate again.
What is concave up?
A curve that bends upward like a valley, where the second derivative is positive.