Calculating Higher-Order Derivatives

A higher-order derivative is obtained by differentiating a function more than once. The first derivative gives the rate of change of the function. The second derivative gives the rate of change of the first derivative (e.g., acceleration in physics), and so on.

Application:

• Second derivative is used to determine concavity and points of inflection.
• Higher-order derivatives appear in Taylor series, physics (jerk, snap), and differential equations.

Notation:

• First derivative: \( f'(x) \) or \( \dfrac{dy}{dx} \)
• Second derivative: \( f”(x) \) or \( \dfrac{d^2y}{dx^2} \)
• Third derivative: \( f^{(3)}(x) \) or \( \dfrac{d^3y}{dx^3} \)
• nth derivative: \( f^{(n)}(x) \) or \( \dfrac{d^ny}{dx^n} \)

Key Idea: Each derivative gives a deeper level of rate-of-change information about the original function.

Examples of Patterns:

• For \( f(x) = e^{kx} \), \( f^{(n)}(x) = k^n e^{kx} \).
• For \( f(x) = \sin(kx) \), derivatives cycle every 4 steps: \( \sin \to \cos \to -\sin \to -\cos \).
• For \( f(x) = \cos(kx) \), similar cycle: \( \cos \to -\sin \to -\cos \to \sin \).