Higher Order Derivatives

Definition

Higher order derivatives are derivatives of derivatives:

Derivative	Notation
First	$f'(x)$, $\frac{dy}{dx}$, $y'$
Second	$f''(x)$, $\frac{d^2y}{dx^2}$, $y''$
Third	$f'''(x)$, $\frac{d^3y}{dx^3}$, $y'''$
Fourth	$f^{(4)}(x)$, $\frac{d^4y}{dx^4}$, $y^{(4)}$
nth	$f^{(n)}(x)$, $\frac{d^ny}{dx^n}$, $y^{(n)}$

Physical Interpretation

If $s(t)$ is position as a function of time:

• $s'(t)$ = velocity (rate of change of position)
• $s''(t)$ = acceleration (rate of change of velocity)
• $s'''(t)$ = jerk (rate of change of acceleration)

Common Patterns

Polynomials

For $f(x) = x^n$:

• $f'(x) = nx^{n-1}$
• $f''(x) = n(n-1)x^{n-2}$
• $f^{(n)}(x) = n!$
• $f^{(n+1)}(x) = 0$

Exponentials

For $f(x) = e^x$:

• $f^{(n)}(x) = e^x$ for all $n$

For $f(x) = e^{ax}$:

• $f^{(n)}(x) = a^n e^{ax}$

Trigonometric Functions

For $f(x) = \sin(x)$:

$f'(x) = \cos(x)$ $f''(x) = -\sin(x)$ $f'''(x) = -\cos(x)$ $f^{(4)}(x) = \sin(x) \quad \text{(cycle repeats)}$

Pattern: $f^{(n)}(x) = \sin\left(x + \frac{n\pi}{2}\right)$

For $f(x) = \cos(x)$:

$f'(x) = -\sin(x)$ $f''(x) = -\cos(x)$ $f'''(x) = \sin(x)$ $f^{(4)}(x) = \cos(x) \quad \text{(cycle repeats)}$

Pattern: $f^{(n)}(x) = \cos\left(x + \frac{n\pi}{2}\right)$

Examples

Example 1: Polynomial

$f(x) = x^5 - 3x^3 + 2x$

$f'(x) = 5x^4 - 9x^2 + 2$ $f''(x) = 20x^3 - 18x$ $f'''(x) = 60x^2 - 18$ $f^{(4)}(x) = 120x$ $f^{(5)}(x) = 120$ $f^{(6)}(x) = 0$

Example 2: Natural log

$f(x) = \ln(x)$

$f'(x) = \frac{1}{x} = x^{-1}$ $f''(x) = -x^{-2} = -\frac{1}{x^2}$ $f'''(x) = 2x^{-3} = \frac{2}{x^3}$ $f^{(4)}(x) = -6x^{-4} = -\frac{6}{x^4}$

Pattern: $f^{(n)}(x) = \frac{(-1)^{n-1}(n-1)!}{x^n}$ for $n \geq 1$

Example 3: Product requiring chain rule

$f(x) = x^2 e^x$

$f'(x) = x^2 e^x + 2xe^x = e^x(x^2 + 2x)$ $f''(x) = e^x(x^2 + 2x) + e^x(2x + 2) = e^x(x^2 + 4x + 2)$ $f'''(x) = e^x(x^2 + 4x + 2) + e^x(2x + 4) = e^x(x^2 + 6x + 6)$

Example 4: Implicit second derivative

For $x^2 + y^2 = 25$, find $\frac{d^2y}{dx^2}$:

First: $\frac{dy}{dx} = -\frac{x}{y}$

$\frac{d^2y}{dx^2} = \frac{d}{dx}\left[-\frac{x}{y}\right]$

$= -\frac{y(1) - x\left(\frac{dy}{dx}\right)}{y^2}$

$= -\frac{y - x\left(-\frac{x}{y}\right)}{y^2}$

$= -\frac{y^2 + x^2}{y^3}$

$= -\frac{25}{y^3}$

The Leibniz Rule

For the nth derivative of a product:

$(fg)^{(n)} = \sum_{k=0}^{n} \binom{n}{k} f^{(n-k)} g^{(k)}$

where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.