Higher-Order Derivatives
Calculate second, third, and higher-order derivatives
Results
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Theory & Formula
Higher order derivatives represent the rate of change of the rate of change. The second derivative measures concavity, the third derivative measures the rate of change of concavity, and so on.
Notation
Multiple notations exist for higher order derivatives:
Second derivative (concavity):
\(f''(x) = \frac{d^2f}{dx^2} = \frac{d}{dx}\left(\frac{df}{dx}\right)\)Third derivative (jerk in physics):
\(f'''(x) = \frac{d^3f}{dx^3}\)nth derivative:
\(f^{(n)}(x) = \frac{d^nf}{dx^n}\)Applications
- Second derivative of position gives acceleration in physics
- Second derivative determines concavity of functions (positive = concave up)
- Third derivative helps find inflection points
- Higher derivatives are used in Taylor series expansions
Example
For f(x) = x⁴:
\(f(x) = x^4\)\(f'(x) = 4x^3\)\(f''(x) = 12x^2\)\(f'''(x) = 24x\)\(f^{(4)}(x) = 24\)\(f^{(5)}(x) = 0\)