The Chain Rule: AP Calculus AB/BC Study Guide
Introduction
Hey there, Calculus Wizards! Ready to dive into the magical world of derivatives within derivatives? Today, we're going to unravel the enigma of the Chain Rule – it's like Inception, but for calculus. 🎢🔗 So grab your thinking cap (or your wizard hat), and let's get started on making differentiation a lot more fun and a little less terrifying!
Composite Functions: Functions within Functions
Composite functions are basically the Russian nesting dolls of math. They're functions within functions. Imagine you have a function f(x) and another function g(x). When you combine them, you get the composite function (f∘g)(x) which is written as f(g(x)). The function g(x) is the "inner function" and f(x) is the "outer function".
For example, let's say:
 ( f(x) = x^2 )
 ( g(x) = 3x + 1 )
The composite function ( f(g(x)) ) is ( f(3x + 1) ). So, ( f(g(x)) = (3x + 1)^2 ). Think of g(x) as the cake batter and f(x) as the oven that bakes it into a delicious cake.
The Chain Rule: Differentiating Composite Functions
The Chain Rule helps us differentiate these composite functions. It's like having a set of instructions for cracking open that nesting doll without breaking it.
There are two notations for the Chain Rule:

( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} )

( \frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x) )
Both basically say: to differentiate ( f(g(x)) ), you first differentiate the outer function ( f ) with respect to the inner function ( g(x) ), and then multiply it by the derivative of the inner function ( g ) with respect to x.
Breaking Down the Chain Rule: Step by Step
To apply the Chain Rule, follow these steps:
 Identify the inner function and the outer function.
 Differentiate the outer function while keeping the inner function intact.
 Differentiate the inner function.
 Multiply the results from steps 2 and 3.
Let’s use this process for some examples:
Example 1: Differentiating ( y = (x^2 + 3x  1)^2 )

Identify Inner and Outer Functions:
 Inner: ( u = x^2 + 3x  1 )
 Outer: ( y = u^2 )

Differentiate Outer Function:
 ( \frac{dy}{du} = 2u )

Differentiate Inner Function:
 ( \frac{du}{dx} = 2x + 3 )

Multiply:
 ( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 2(x^2 + 3x  1) \cdot (2x + 3) )
Example 2: Differentiating ( y = e^{\ln(x)} )

Identify Inner and Outer Functions:
 Inner: ( g(x) = \ln(x) )
 Outer: ( f(x) = e^u )

Differentiate Outer Function:
 ( f'(g(x)) = e^{g(x)} = e^{\ln(x)} )

Differentiate Inner Function:
 ( g'(x) = \frac{1}{x} )

Multiply:
 ( \frac{dy}{dx} = e^{\ln(x)} \cdot \frac{1}{x} = x \cdot \frac{1}{x} = 1 )
Example 3: Differentiating ( y = 4(5x^3 + 2x^2 + 6)^2 )

Identify Inner and Outer Functions:
 Inner: ( u = 5x^3 + 2x^2 + 6 )
 Outer: ( y = 4u^2 )

Differentiate Outer Function:
 ( \frac{dy}{du} = 8u )

Differentiate Inner Function:
 ( \frac{du}{dx} = 15x^2 + 4x )

Multiply:
 ( \frac{dy}{dx} = 8(5x^3 + 2x^2 + 6) \cdot (15x^2 + 4x) )
Example 4: Differentiating ( y = \sqrt{7x^2} )

Identify Inner and Outer Functions:
 Inner: ( u = 7x^2 )
 Outer: ( y = u^{1/2} )

Differentiate Outer Function:
 ( \frac{dy}{du} = \frac{1}{2}u^{1/2} )

Differentiate Inner Function:
 ( \frac{du}{dx} = 14x )

Multiply:
 ( \frac{dy}{dx} = \frac{1}{2}(7x^2)^{1/2} \cdot 14x = \frac{1}{2}\cdot \frac{1}{\sqrt{7x^2}} \cdot 14x = \frac{7x}{\sqrt{7x^2}} = \frac{7x}{\sqrt{7 \cdot x^2}} = \frac{7x}{x\sqrt{7}} = \frac{7}{\sqrt{7}} = \sqrt{7} )
Practice Problem: 🧠
Now it's your turn! Find the derivative of ( f(x) = \cos^2(3x) ).
 Hint: Inner function ( u = \cos(3x) ), outer function ( f(x) = u^2 )
 Answer: ( f'(x) = 2(\cos(3x)) \cdot (3\sin(3x)) = 6\cos(3x)\sin(3x) )
Closing
And there you have it, Calculus Crusaders! You've now mastered the Chain Rule. Remember, practice makes perfect. Keep tackling those problems and soon you’ll be chaining those derivatives like a pro! Who said calculus couldn’t be entertaining? Keep those pencils sharp, those brains sharper, and may the function always be in your favor. 🌟🧙♂️