#Unit 3: Differentiation - Mastering the Chain Rule 🚀

Welcome to Unit 3! This unit is all about mastering different differentiation techniques, and we're kicking it off with the Chain Rule, a fundamental concept for handling composite functions. Let's dive in!

#🔄 Composite Functions

Given two functions, $f(x)$ and $g(x)$, the composite function $(f \circ g)(x)$ is formed by applying $f$ to the output of $g$. Mathematically:

$$(f \circ g)(x) = f(g(x))$$

• $g(x)$ is the inner function.
• $f(x)$ is the outer function.

#Composite Function Example

Let's break it down with an example:

$$\textcolor{red}{f(x) = x^2}$$

$$\textcolor{blue}{g(x) = 3x + 1}$$

For $f(g(x))$, we have:

• The inner function, $g(x)$, takes $x$, multiplies it by 3, and adds 1. - The outer function, $f(x)$, takes the result from $g(x)$ and squares it.

Therefore:

$$f(g(x)) = \textcolor{red}{(\textcolor{blue}{3x + 1})^2}$$

Understanding this is key to using the Chain Rule!

#🔗 Definition of The Chain Rule

The Chain Rule is your go-to method for differentiating composite functions. It's used a lot, so make sure you know it well!

There are two common notations:

1. Leibniz Notation:

   $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

   • $\frac{dy}{dx}$ is the overall derivative.
   • $u$ is the ...