Unit 3 · Topic 3.5

3.5 Selecting Procedures for Calculating Derivatives

Real derivative problems mix rules: a chain rule inside a product rule, a quotient of composites, and so on. The skill is reading the structure of the function, choosing rules in the right order and rewriting first when it makes the work easier.

Key terms

combining derivative rules
rewriting before differentiating
order of operations
strategy

Read the structure first

Before writing anything, ask what the last operation in the function is. That tells you the first rule to use.

Then repeat the question for each piece. If you're using the product rule on x³ sin(2x), the second factor, sin(2x), needs the chain rule when you differentiate it.

Last operation	First rule
Addition or subtraction	Sum or difference rule
Multiplication of two functions of x	Product rule
Division by a function of x	Quotient rule (or rewrite)
A function applied to an expression	Chain rule

Rewrite before you differentiate

A minute of rewriting can save many lines of algebra. Common moves:

Constant over an expression: 5/(2x + 1)³ = 5(2x + 1)⁻³. Use the chain rule instead of the quotient rule.
Roots: √(x² + 4) = (x² + 4)^(1/2).
Logarithms of products, quotients and powers: ln(x√(x + 1)) = ln x + ½ ln(x + 1). Log properties turn one hard derivative into easy ones.
Single-term denominators: (x² + 3x)/√x = x^(3/2) + 3x^(1/2).

Functions given by tables and graphs

When the functions aren't formulas, the same rules apply, but you plug in values instead of expressions. Write the derivative in general form first, like h′(x) = f′(x²)·2x·g(x) + f(x²)·g′(x), then substitute the x-value and look up each piece. Inner functions change which row of the table you need.

The exam often uses names other than f and g, like p, q or w. The rules don't change.

A checklist for long derivatives

For a long function, set up the skeleton before you fill in details:

Name the outermost operation and write that rule's template, like (first)′(second) + (first)(second)′.
Differentiate each piece separately on scratch paper, using the chain rule where a piece has an inside.
Fill the pieces into the template.
Simplify only if the question needs it, like solving f′(x) = 0 or finding a sign.

Checking your work

On calculator-active questions, compare your formula's value at one point with the calculator's numerical derivative. If they agree to several decimal places, your formula is almost certainly right. On no-calculator questions, a quick sense check of signs (is the function rising or falling there?) catches many errors.

Worked examples

Try each one yourself first, then open the solution.

Example 1
Product rule with a chain rule inside
Differentiate y = x³ sin(2x).
Show the solutionHide the solution
1. Step 1: Last operation: multiplication, so start with the product rule.
2. Step 2: Derivative of x³ is 3x². Derivative of sin(2x) is cos(2x)·2 by the chain rule.
3. Step 3: y′ = 3x² sin(2x) + x³·2 cos(2x).
Answer: y′ = 3x² sin(2x) + 2x³ cos(2x)
Example 2
Rewrite a log first
Differentiate y = ln(x√(x + 1)) for x > 0.
Show the solutionHide the solution
1. Step 1: Use log properties: ln(x√(x + 1)) = ln x + ln((x + 1)^(1/2)) = ln x + ½ ln(x + 1).
2. Step 2: Differentiate each term: 1/x + ½·1/(x + 1).
3. Step 3: Combine if you like: (2(x + 1) + x)/(2x(x + 1)) = (3x + 2)/(2x(x + 1)).
Answer: y′ = 1/x + 1/(2(x + 1)) = (3x + 2)/(2x(x + 1))
Example 3
Trap: a composite inside a product, from a table
Values: f(2) = 6, f′(2) = 1, f(4) = −1, f′(4) = 3, g(2) = 5, g′(2) = 2. If h(x) = f(x²)·g(x), find h′(2).
Show the solutionHide the solution
1. Step 1: Product rule, with the chain rule on f(x²): h′(x) = f′(x²)·2x·g(x) + f(x²)·g′(x).
2. Step 2: At x = 2, the inner value is x² = 4. So you need f′(4) and f(4), not f′(2) and f(2).
3. Step 3: h′(2) = f′(4)·4·g(2) + f(4)·g′(2) = 3·4·5 + (−1)·2.
4. Step 4: = 60 − 2 = 58.
Answer: h′(2) = 58

Common mistakes

Applying rules in the wrong order, such as using the chain rule on the whole product x³ sin(2x).
Using the input x instead of the inner output (like x²) when reading a table.
Splitting ln(x + 1) into ln x + ln 1. Log rules split products, not sums.

On the exam

Mixed-rule derivatives, especially from tables, are among the most common multiple-choice questions.
In free response, you can leave derivatives unsimplified unless you need to analyze them further. Just make sure each rule is clearly applied.

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