Derivative Definition – AP Calculus AB

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

Limit Mechanism

Distance ($\Delta x$) = 3.00

The Formal Definition

$f'(x) = \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}$

CALCULATIONS AT X = 0.5

Secant Slope ($m_{sec}$) 1.25

Tangent Slope ($m_{tan}$) -0.81

Error (Difference): 2.06

Observation: As you drag $\Delta x$ closer to 0, the secondary point $f(x+\Delta x)$ slides down the curve. The purple secant line rotates until it PERFECTLY aligns with the green tangent line.

AP Exam Tip

You MUST be able to recognize the limit definition when it's disguised. For example, if you see $\lim_{h \to 0} \frac{\sin(\pi+h)-\sin(\pi)}{h}$, instantly recognize that it's just asking for the derivative of $\sin(x)$ evaluated at $x=\pi$, which is $\cos(\pi) = -1$!