The Power of Derivatives

Derivatives are all about change! They measure how a quantity changes as something else changes. In math terms, the derivative of a function gives you the slope of the tangent line at any point.

Notation and Definition

The derivative of \( f(x) \) at \( x = a \) is written as \( f'(a) \) or \( \frac{df}{dx} \bigg|_{x=a} \), and is defined by:

\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]

What Derivatives Tell Us

• If \( f'(x) > 0 \), the function is increasing.
• If \( f'(x) < 0 \), the function is decreasing.
• If \( f'(x) = 0 \), the function might have a maximum, minimum, or a “flat spot.”

Real-Life Meaning

Derivatives help us find velocities, growth rates, and any situation where “rate of change” is important.

Basic Rules

• Power rule: \( \frac{d}{dx} x^n = n x^{n-1} \)
• Sum rule: The derivative of a sum is the sum of the derivatives.