AcademyDifferential Calculus
Academy
Higher Derivatives
Level 1 - Calculus topic page in Differential Calculus.
Principle
Higher derivatives repeat the process of differentiation. They measure how rates of change themselves change. For example, the second derivative measures curvature and acceleration.
Notation
\(f''(x)\)
second derivative of f
\(f^{(n)}(x)\)
nth derivative of f
\(\frac{d^2y}{dx^2}\)
second derivative of y with respect to x
The Core Method
Differentiate once to get \(f'\), then differentiate the result to get \(f''\), and continue as needed.
Second derivative
\[f''(x)=\frac{d}{dx}\left(f'(x)\right)\]
Keep notation clear: \(f^2(x)\) means \((f(x))^2\), but \(f''(x)\) means the second derivative.
Worked Cases
Question
For \(f(x)=x^4-3x^2+2\), find \(f''(x)\).
Answer
First differentiate: \(f'(x)=4x^3-6x\). Differentiate again: \(f''(x)=12x^2-6\).
Examples
Question
Find \(f^{(3)}(x)\) for \(f(x)=x^5\).
Answer
Differentiate repeatedly: \(f'(x)=5x^4\), \(f''(x)=20x^3\), and \(f^{(3)}(x)=60x^2\).