Topic 3.5 Notes – Selecting Procedures for Calculating Derivatives

What Selecting a Procedure Actually Means

When you look at a function, don’t start differentiating immediately. First ask:

What is the outermost operation?

Every derivative problem is built from:

• sums or differences
• products
• quotients
• compositions (a function inside another function)

Your first job is to identify the outside structure, because that determines the main rule.

After that, you move inward and apply additional rules as needed.

The Derivative Rules You’re Choosing From

You already know these. Now we organize them by when they apply.

Basic Rules (supporting rules)

• Power rule: $\frac{d}{dx}x^n = nx^{n-1}$
• Constant rule
• Constant multiple rule
• Sum/difference rule

These usually happen automatically inside bigger rules.

Product Rule

Used when two or more functions are multiplied.

$$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$

Clue: separate factors multiplied together.

Example structure:
$x^2 \sin x$

That’s two functions multiplied → product rule.

If you see three factors like $x e^x \cos x$, you apply the product rule twice.

Quotient Rule

Used when one function is divided by another.

$$\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{f'g - fg'}{g^2}$$

Clue: a function in the numerator and a function in the denominator.

Example structure:
$\frac{\ln x}{x^2+1}$

That’s clearly a quotient.

Quick insight: sometimes rewriting helps.
For example,

$$\frac{3}{x^2} = 3x^{-2}$$

Now you can just use the power rule.

On a no-calculator MCQ, rewriting to avoid the quotient rule can save time.

Chain Rule

Used for composite functions.

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

Clue: something is inside something else.

Examples:

• $(4x-1)^6$
• $\cos(x^3)$
• $e^{5x^2}$

Think in layers:

• Outside function
• Inside function
• Derivative of outside (leave inside alone)
• Multiply by derivative of inside

If you forget the chain rule, you will lose easy points.

How to Choose the Right Rule

Think from the outside moving inward.

Many AP problems are built in layers.

Example 1

$$f(x) = \frac{\sin(2x)}{x^3}$$

Outermost structure → quotient.
Inside numerator → composite (chain rule).
Denominator → power rule.

So the order is:

1. Quotient rule
2. Chain rule inside the numerator
3. Power rule where needed

Example 2

$$g(x) = (x^2+1)^4 \cos x$$

Outermost structure → product.
Inside first factor → composite.

Order:

1. Product rule
2. Chain rule on $(x^2+1)^4$

Students often try to expand first. That’s messy and unnecessary.

Common Mistakes I See Every Year

Mixing Up Product and Chain

$$(x^2+3)^5$$

This is NOT a product.
It is one expression raised to a power → chain rule.

If there is no multiplication outside the parentheses, it’s not a product.

Forgetting to Apply Multiple Rules

If you see:

• product + composite
• quotient + composite
• product of three functions

You must apply rules in layers.

AP multiple choice often tests whether you recognize the sequence, not just compute.

Sign Errors in the Quotient Rule

The numerator is $f'g - fg'$.
Students flip the order constantly.

On FRQs, that small sign error can cost multiple points.

When to Simplify First

Before differentiating, ask:

• Can I rewrite a radical as a power?
• Can I rewrite a quotient with negative exponents?
• Is expanding easier than product rule?

Example:

$$h(x) = x(x^2 - 5)$$

Expanding first gives $x^3 - 5x$.
Now it’s just the power rule.

Strategic simplification is part of selecting the right procedure.

How This Shows Up on the AP Exam

• Multiple choice often asks which sequence of rules applies.
• FRQs expect clean structure before simplification.
• If you choose the wrong rule, everything after is wrong even if algebra is perfect.

The exam rewards recognizing structure quickly.