Selecting Procedures for Calculating Derivatives

AP Calculus · Differentiation: Composite, Implicit, and Inverse Functions

Selecting Procedures for Calculating Derivatives

No new rules. The skill is naming the outermost operation before you differentiate anything.

key idea$\text{Read the STRUCTURE before you differentiate anything}$

How to choose

No new technique appears here. What is tested is whether you can look at an expression and name its top-level structure — because the outermost operation decides which rule leads, and everything else nests inside it.

Step	What you do	Why it comes first
1	Ask: can this be simplified first?	expanding or dividing often removes the need for any rule
2	Identify the last operation you would perform to evaluate it	that operation is the outermost, and its rule leads
3	Apply that rule, treating the parts as black boxes	keeps the structure intact while you work
4	Recurse into each part	inner structures get their own rules

Step 2 is the whole skill. “What would I do last if I were plugging in a number?” identifies the outer operation every time.

The decision tree

Simplify if you can. Then find the outermost operation below.

SIMPLIFY?

The expression divides, expands or cancels.Do that first. $\dfrac{x^{3}+2x}{x}$ is $x^{2}+2$; $x^{2}\cdot x^{5}$ is $x^{7}$. A rule applied to an unsimplified expression is correct but slow and error-prone.

OUTER = $+$ or $-$

Terms joined at the top level.Split. Differentiate each term independently — linearity. Then recurse into each one.

OUTER = $\times$

Two functions multiplied, neither constant.Product rule. $f'g+fg'$. If one factor is a constant, it is a constant multiple instead, not a product.

OUTER = $\div$

One function over another, denominator not constant.Quotient rule — or rewrite as a product with a negative exponent if that is cleaner.

OUTER = composition

A function inside another: brackets, roots, exponents, trig arguments.Chain rule. Differentiate outer, keep the inside, multiply by the inner derivative.

OUTER = an equation in $x$ and $y$

$y$ cannot be isolated.Implicit differentiation. Differentiate both sides; every $y$ carries $\tfrac{dy}{dx}$.

Expression	Outermost operation	Leading rule
$x^{2}\sin x$	multiplication	product
$\sin\left(x^{2}\right)$	composition	chain
$\dfrac{\sin x}{x^{2}}$	division	quotient
$x^{2}+\sin x$	addition	split, then basic rules
$\left(x^{2}\sin x\right)^{4}$	composition	chain, then product inside
$\dfrac{x^{2}+x}{x}$	—	simplify first → $x+1$

Rows 1–3 use the same two functions and need three different rules. The operation joining them is the only thing that changed.

Worked examples

EX 1

Same pieces, three different rules

discrimination

Differentiate $x^{2}\sin x$, $\sin\left(x^{2}\right)$ and $\dfrac{x^{2}}{\sin x}$.

First: multiplication

Product rule: $2x\sin x+x^{2}\cos x$.

Second: composition

Chain rule with inner $x^{2}$: $\cos\left(x^{2}\right)\cdot2x=2x\cos\left(x^{2}\right)$.

Third: division

Quotient rule: $\dfrac{2x\sin x-x^{2}\cos x}{\sin^{2}x}$.

Compare

Identical ingredients, three unrelated answers. The joining operation did all the work.

The habit

Before differentiating, say out loud which operation is outermost. That one sentence prevents most errors on this topic.

EX 2

Nested structures

$\left(x^{2}\sin x\right)^{4}$

Differentiate $y=\left(x^{2}\sin x\right)^{4}$.

Outermost first

The last operation is raising to the 4th power — a composition, so the chain rule leads.

Apply it, treating the inside as a black box

$y'=4\left(x^{2}\sin x\right)^{3}\cdot\dfrac{d}{dx}\left[x^{2}\sin x\right]$.

Now recurse

The inside is a product: $\dfrac{d}{dx}\left[x^{2}\sin x\right]=2x\sin x+x^{2}\cos x$.

Assemble

$y'=4\left(x^{2}\sin x\right)^{3}\left(2x\sin x+x^{2}\cos x\right)$.

Note the order

Chain outside, product inside. Reversing them produces nonsense.

EX 3

The one that needs no rule

$\dfrac{x^{3}-x}{x}$

Differentiate $y=\dfrac{x^{3}-x}{x}$.

Check for simplification

The denominator is a single power and the numerator is a sum, so division distributes.

Simplify

$y=x^{2}-1$ for $x\neq0$.

Differentiate

$y'=2x$.

Compare the alternative

The quotient rule gives $\dfrac{\left(3x^{2}-1\right)x-\left(x^{3}-x\right)}{x^{2}}=\dfrac{2x^{3}}{x^{2}}=2x$ — same answer, four times the work.

Why step 1 exists

Checking for a simplification costs five seconds and sometimes saves the entire problem.

EX 4

Constant multiple, not a product

$5\cos x$

A student applies the product rule to $y=5\cos x$, getting $0\cdot\cos x+5\cdot(-\sin x)$. Comment.

Check the answer

It simplifies to $-5\sin x$, which is correct.

Check the classification

$5$ is a constant, not a function of $x$, so this is a constant multiple, not a product.

Note what the product rule did

It handled the case correctly because $\tfrac{d}{dx}[5]=0$ kills the extra term. The rule is not wrong, just unnecessary.

Give the direct route

Constant multiple rule: $5\cdot\dfrac{d}{dx}[\cos x]=-5\sin x$.

The discriminator

A product needs both factors to depend on $x$. If one is a number, pull it out and move on.

EX 5

When $y$ will not isolate

$x^{2}y+\sin y=4$

Which technique differentiates $x^{2}y+\sin y=4$, and what is the first line?

Try to isolate $y$

$x^{2}y+\sin y=4$ mixes $y$ algebraically and inside a trig function. No rearrangement isolates it.

Choose the technique

Implicit differentiation — differentiate both sides as they stand.

Handle each term

$x^{2}y$ is a product: $2xy+x^{2}\dfrac{dy}{dx}$. $\sin y$ is a composition: $\cos y\dfrac{dy}{dx}$. The constant gives $0$.

Write the first line

$2xy+x^{2}\dfrac{dy}{dx}+\cos y\dfrac{dy}{dx}=0$.

Note the nesting

Implicit differentiation is the outer decision; product and chain rules operate inside it. The routing question is answered before any rule is applied.

Watch out

Differentiating before reading the structure

Most errors on this topic are a correct rule applied to the wrong level. Name the outermost operation first, every time.

Skipping the simplification check

$\dfrac{x^{3}-x}{x}$ and $x^{2}\cdot x^{5}$ need no rule at all. Five seconds of inspection saves minutes.

A constant factor is not a product

$5\cos x$ needs the constant multiple rule. Using the product rule works but signals you have not read the structure.

Rules nest, they do not compete

$\left(x^{2}\sin x\right)^{4}$ needs chain and product, in that order. Asking “which rule?” as though only one applies is the wrong question.

Course map. Topic 3.5 · The CED assigns this topic no unique learning objective — it is a synthesis of the differentiation rules, assessed as procedure selection.