Lesson 5: Basic Derivative Rules (Power, Constant, Sum)

1. Constant Rule

The derivative of any constant is zero.

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

Example: \(d/dx [7] = 0, d/dx [-5.3] = 0\)

Intuition: a horizontal line has slope 0 everywhere.

2. Power Rule

Bring down the exponent and subtract one from it:

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

Works for any real \(n\) (positive, negative, fractions).

Examples:

• \(d/dx [x^4] = 4x^3\)
• \(d/dx [x^{-2}] = -2x^{-3}\)
• \(d/dx [√x] = d/dx [x^{1/2}] = (1/2) x^{-1/2} = 1/(2√x)\)

3. Constant Multiple & Sum/Difference Rules

\(d/dx [c · f(x)] = c · f'(x)\)

\(d/dx [f(x) ± g(x)] = f'(x) ± g'(x)\)

Example: \(d/dx [5x^3 - 2x^2 + 9x - 4] = 15x^2 - 4x + 9\)