#AP Calculus AB/BC: Derivatives - The Ultimate Study Guide 🚀

Hey there, future calculus master! This guide is your go-to resource for acing the derivatives section of the AP Calculus exam. Let's break down everything you need to know, making it super clear and easy to remember. Time is precious, so let's get right to it!

This unit is crucial! It makes up 10-12% of the Calculus AB exam and 4-7% of the Calculus BC exam. Nail this, and you're setting yourself up for success. 💪

#1. What is a Derivative? 🤔

#1.1 The Idea of Instantaneous Change

Imagine Michael Phelps swimming. His average speed is easy to calculate, but what about his speed at a single moment? That's where derivatives come in!

We use a function to represent his position over time. The slope of this function over an interval gives us average speed. To find his speed at a specific instant, we shrink that interval down to almost zero. This process is the essence of a derivative.

#1.2 Formal Definition

Let's get a bit formal. If we have a function s(t) representing position over time, the derivative (instantaneous speed) is:

$$\frac{ds}{dt} = \lim_{\Delta t \to 0} \frac{s(t + \Delta t) - s(t)}{\Delta t}$$

Here, ds/dt is the derivative of s with respect to t. The d notation indicates we're dealing with infinitesimally small changes—derivatives!

In the xy-plane, for a function f(x), the derivative is:

$$\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$

This can also be written as f'(x) or y'. While these notations are useful, dy/dx helps you remember that you're finding the rate of change of y with resp...