Basic Differentiation Rules

Constant Rule

The derivative of a constant is zero:

$\frac{d}{dx}[c] = 0$

Power Rule

For any real number $n$:

$\frac{d}{dx}[x^n] = n \cdot x^{n-1}$

Examples

• $\frac{d}{dx}[x^5] = 5x^4$
• $\frac{d}{dx}[x^{-2}] = -2x^{-3}$
• $\frac{d}{dx}[\sqrt{x}] = \frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$

Constant Multiple Rule

$\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$

Example

$\frac{d}{dx}[5x^3] = 5 \cdot (3x^2) = 15x^2$

Sum Rule

$\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$

Difference Rule

$\frac{d}{dx}[f(x) - g(x)] = f'(x) - g'(x)$

Common Derivatives Table

Function	Derivative
$c$ (constant)	$0$
$x$	$1$
$x^n$	$n \cdot x^{n-1}$
$\sqrt{x}$	$\frac{1}{2\sqrt{x}}$
$\frac{1}{x}$	$-\frac{1}{x^2}$
$\frac{1}{x^n}$	$-\frac{n}{x^{n+1}}$

Polynomial Derivatives

For a polynomial $p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$:

$p'(x) = n \cdot a_n x^{n-1} + (n-1) \cdot a_{n-1} x^{n-2} + \cdots + a_1$

Examples

Example 1: Polynomial

$f(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 4$

$f'(x) = 12x^3 - 6x^2 + 10x - 7$

Example 2: Rational exponents

$f(x) = x^{3/2} + x^{-1/3}$

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

Example 3: Rewriting before differentiating

$f(x) = \frac{x^3 + 2x}{x^2} = x + 2x^{-1}$

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

Example 4: Finding slope at a point

Find the slope of $y = x^4 - 3x^2$ at $x = 2$:

$y' = 4x^3 - 6x$

$y'(2) = 4(8) - 6(2) = 32 - 12 = 20$