AcademyDifferentiation
Academy
Higher Order Derivatives
Level 1 - Math I (Physics) topic page in Differentiation.
Higher Order Derivatives
The second derivative is the derivative of the first derivative.
Second Derivative
\[f''(x) = \frac{d}{dx}[f'(x)]\]
Notation
- First derivative: \(f'(x) = \frac{dy}{dx}\)
- Second derivative: \(f''(x) = \frac{d^2y}{dx^2}\)
- Third derivative: \(f'''(x) = \frac{d^3y}{dx^3}\)
Physical Interpretation
For position \(s(t)\):
- First derivative \(s'(t)\) = velocity \(v(t)\)
- Second derivative \(s''(t)\) = acceleration \(a(t)\)
Example
If \(f(x) = x^3\):
Example
\[f'(x) = 3x^2, \quad f''(x) = 6x, \quad f'''(x) = 6\]