The Second Derivative
The second derivative \(f''(x)\) measures how the gradient itself changes, giving concavity and, in motion, acceleration.
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 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.