AP Precalculus: Introduction to Derivatives – Core Formulas

1. Average Rate of Change

• \[ \text{Average rate of change on } [a, b]:\ \frac{f(b)-f(a)}{b-a} \]
• This is the slope of the secant line through points \( (a, f(a)) \) and \( (b, f(b)) \)
• Velocity as average rate: \( v_{\text{avg}} = \frac{s(t_2)-s(t_1)}{t_2-t_1} \) for position function \( s(t) \)

2. Instantaneous Rate of Change / Definition of the Derivative

• \[ f'(a) = \lim_{h\to 0} \frac{f(a+h) - f(a)}{h} \]
• This gives the **slope of the tangent line** to \( f(x) \) at \( x = a \)
• Also called the **instantaneous rate of change** at \( x=a \)

3. Velocity as a Rate of Change

• If \( s(t) \) is position at time \( t \), then **instantaneous velocity** at \( t=a \) is: \[ v(a) = s'(a) = \lim_{h\to 0} \frac{s(a+h) - s(a)}{h} \]

4. Finding Derivatives/Slope with Limits

• For \( f(x) \), **derivative at \( a \)** using limit: \[ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x-a} \]
• Slope of tangent is just \( f'(a) \)

5. Equation of the Tangent Line

• At \( x=a \): \[ y = f(a) + f'(a)(x - a) \]
• This line passes through \( (a, f(a)) \) with slope \( f'(a) \)