Topic 3.5 Notes – Selecting Procedures for Calculating Derivatives

Recognizing the Structure of a Function

Before taking any derivative, pause and ask: What kind of function is this on the outside? Almost every problem falls into one (or a mix) of these categories.

1. Basic forms

These don’t require special structure rules.

• Polynomials → Power Rule
• Exponential and logarithmic functions
• Trig functions
• Inverse trig functions

Example:
If $f(x) = 7x^4 - 2x$, you’re just using the power rule term by term.

2. Sums and Constant Multiples

Derivatives distribute nicely:

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

These are the easiest parts of messy problems. Don’t overthink them.

3. Products

If two or more non-constant expressions are multiplied, you need the Product Rule:

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

If there are three factors, apply the product rule twice.

Example:
$h(x) = x^2 \sin x$
Two factors → Product Rule.

4. Quotients

If one function is divided by another, use the Quotient Rule:

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

Sometimes rewriting helps.
Example:
$$\frac{3x^2}{x^5} = 3x^{-3}$$
Power rule is much easier than quotient rule here.

Strong students look for simplifications first.

5. Compositions (Chain Rule)

If one function is inside another, use the Chain Rule:

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

You’ll see this when:

• Something is inside parentheses
• Something is in an exponent
• Something is inside a trig function
• Something is under a radical

Example:
$f(x) = \sin(4x^3)$

Outer function → sine
Inner function → $4x^3$

Derivative:
$$\cos(4x^3)\cdot 12x^2$$

That inside derivative is the most common thing students forget on quizzes.

6. Implicitly Defined Relationships

If x and y are mixed together, like:

$$x^2 + y^2 = 25$$

You use implicit differentiation.

Differentiate both sides with respect to x.
Whenever you differentiate a y-term, multiply by $\frac{dy}{dx}$.

Example step:
$$\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}$$

That extra $\frac{dy}{dx}$ comes from the chain rule. On FRQs, missing it costs points instantly.

Deciding Which Rule Comes First

This is the heart of the topic.

Always identify the outermost structure first.

Let’s look at an example:

$$f(x) = \frac{e^{x^2}}{x^3 + 1}$$

What’s happening on the outside? It’s a quotient.

So:

1. Start with the Quotient Rule.
2. When differentiating $e^{x^2}$, use the Chain Rule.
3. When differentiating $x^3+1$, use the Power Rule.

Outer rule first. Inner rules second.

Here’s another:

$$g(x) = x^2 \cos(3x)$$

Outer structure → product.
So use Product Rule first.

Inside that:

• Derivative of $x^2$ → power rule
• Derivative of $\cos(3x)$ → chain rule

Think in layers.

When Multiple Rules Combine

AP questions love combinations like:

• Product + Chain
• Quotient + Chain
• Implicit + Product
• Product used twice

If you see trig, exponentials, or logs with something inside them, the chain rule is almost guaranteed to appear somewhere.

If you see three multiplied factors, expect product rule twice.

On multiple-choice, many wrong answers come from:

• Using the right rule in the wrong order
• Forgetting the inner derivative
• Messing up signs in the quotient rule

The structure determines everything.

Smart Rewrites Before Differentiating

Before diving in, ask:

• Can I rewrite a quotient with negative exponents?
• Can I factor out constants?
• Can I simplify algebra first?

Example:
$$\frac{5x^4 + 10x^2}{5}$$

Simplify first → $x^4 + 2x^2$.
Now it’s just power rule.

That saves time and reduces mistakes.