Topic 3.1 Notes – The Chain Rule

1. Composite Functions and What the Chain Rule Says

A composite function happens when the output of one function becomes the input of another.

If
$$y = f(g(x)),$$

• $g(x)$ is the inner function
• $f(\,\cdot\,)$ is the outer function
• The whole expression is “$f$ of $g(x)$”

Here’s the structure visually.

Function composition diagram

The derivative rule for this situation is the Chain Rule.

The Chain Rule formula

If $y = f(g(x))$, then

$$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$$

Another common notation is

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

Both say the same thing:

• Differentiate the outer function
• Keep the inside unchanged
• Multiply by the derivative of the inner function

Think: outer derivative × inner derivative

2. How to Apply the Chain Rule

Any time you see “something inside something,” your Chain Rule radar should go off.

Let’s walk through the logic clearly.

Step 1 Identify the layers

Ask yourself:

• What operation is happening last?
• What expression is being treated as a single unit?

For example:

• $(3x^2 - 5x + 4)^7$
  Outer = power of 7
  Inner = $3x^2 - 5x + 4$
• $\sin(4x^3)$
  Outer = sine
  Inner = $4x^3$

Step 2 Differentiate the outer function

Pretend the inner expression is just a variable like $u$.

Examples:

• $(u)^7 \to 7u^6$
• $\ln(u) \to \frac{1}{u}$
• $e^u \to e^u$
• $\cos(u) \to -\sin(u)$

Do not touch the inside yet.

Step 3 Multiply by the derivative of the inner function

Now differentiate the inside with respect to $x$, and multiply.

Quick Example

Differentiate
$$y = (5x^3 - 2x)^4$$

• Outer derivative: $4(5x^3 - 2x)^3$
• Inner derivative: $15x^2 - 2$

Final answer:

$$\frac{dy}{dx} = 4(5x^3 - 2x)^3(15x^2 - 2)$$

Notice the inside stayed intact until the multiplication step.

3. Common Chain Rule Structures on AP Exams

These patterns show up constantly on quizzes, FRQs, and both calculator and no-calculator MCQs.

a. Power of a polynomial

$$(g(x))^n$$

Derivative:

$$n(g(x))^{n-1} \cdot g'(x)$$

This is the most common Chain Rule trigger.

b. Exponentials

$$e^{g(x)}$$

Derivative:

$$e^{g(x)} \cdot g'(x)$$

The exponential part stays the same. Students often forget the $g'(x)$.

c. Natural logarithms

$$\ln(g(x))$$

Derivative:

$$\frac{1}{g(x)} \cdot g'(x)$$

On FRQs, this often appears inside bigger expressions.

d. Trig functions

$$\sin(g(x)), \quad \cos(g(x)), \quad \tan(g(x))$$

Example:

$$\frac{d}{dx}[\sin(7x^2)] = \cos(7x^2)\cdot 14x$$

You must multiply by the derivative of the angle.

e. Radicals

Rewrite first:

$$\sqrt{g(x)} = (g(x))^{1/2}$$

Then apply power rule + Chain Rule.

Multiple layers

If you see something like
$$\cos^3(2x)$$

There are three layers:

1. Power of 3
2. Cosine
3. $2x$

You take three derivatives and multiply them all.

This is where students miss a factor.

4. When to Use It and Common Mistakes

When you need the Chain Rule

• A function raised to a power and the base is not just $x$
• A trig/log/exponential function with more than just $x$ inside
• Any clear “function inside a function”

If parentheses are doing real work, you probably need it.

Common mistakes

• Forgetting the inner derivative
  This is the number one lost point on tests.
• Differentiating the inside too soon
  Outer first. Always.
• Missing layers
  Especially with trig powers like $\sin^4(3x)$.
• Dropping parentheses
  Keep the inside grouped until the end.

On the AP exam, many derivatives look simple but are layered. They test whether you truly see structure.