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Selecting Procedures for
Calculating Derivatives

Master the art of choosing the right differentiation technique for any function: power rule, product rule, quotient rule, and chain rule.

AP Calculus AB

Why Procedure Selection Matters

Calculus gives you a toolkit of differentiation rules. The challenge is not just knowing each rule: it is recognizing which rule to apply when you see a new function. The wrong choice leads to unnecessary complexity or outright errors.

The Derivative Toolkit

Power Rule
\(\frac{d}{dx}[x^n] = nx^{n-1}\)
Product Rule
\(\frac{d}{dx}[f \cdot g] = f'g + fg'\)
Quotient Rule
\(\frac{d}{dx}\!\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}\)
Chain Rule
\(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)

Recognizing Function Structure

Identify the Operation Sum / Difference Product / Quotient Composition Differentiate Each Term Product Rule Quotient Rule Chain Rule
\(3x^2 + \sin x\)
\(x^2 \cdot \cos x\)
\(\frac{\sin x}{x^2}\)
\(\sin(x^2)\)

The Power Rule in Depth

The power rule applies to any real exponent: positive, negative, or fractional. Before differentiating, rewrite radicals and reciprocals as powers:

\(\sqrt{x}\)
Rewrite as \(x^{1/2}\), then derivative is \(\frac{1}{2}x^{-1/2}\)
\(\dfrac{1}{x^3}\)
Rewrite as \(x^{-3}\), then derivative is \(-3x^{-4}\)

The Product Rule

Use the product rule whenever two non-constant functions are multiplied together. The key signal is a multiplication between two expressions that both depend on \(x\).

1
Identify the two factors: \(f(x)\) and \(g(x)\).
2
Differentiate each factor separately: find \(f'(x)\) and \(g'(x)\).
3
Combine: \(f'g + fg'\). Don't forget to simplify.

The Quotient Rule

Use the quotient rule when one function is divided by another, and the denominator contains \(x\). The mnemonic "lo d-hi minus hi d-lo, over lo-lo" captures the pattern.

The Chain Rule

x Inner g(x) Outer f(u) y Multiply by g'(x) Differentiate f Derivative = f'(g(x)) · g'(x)

Product vs Quotient Rule

Product Rule Quotient Rule
Structure \(f(x) \cdot g(x)\) \(\dfrac{f(x)}{g(x)}\)
Formula \(f'g + fg'\) \(\dfrac{f'g - fg'}{g^2}\)
Key Sign + (addition) - (subtraction)
Denominator? None \([g(x)]^2\)
Example \(x^2 \sin x\) \(\dfrac{\sin x}{x^2}\)

The Decision Flowchart

1
Simplify first: Rewrite radicals as fractional powers, distribute constants, and combine like terms if possible.
2
Identify the outermost operation: Is it a sum, product, quotient, or composition?
3
Select the matching rule: Sum → term-by-term. Product → product rule. Quotient → quotient rule. Composition → chain rule.
4
Apply recursively: Each sub-expression may itself require another rule. Work from the outside in.
5
Simplify the result: Factor, combine fractions, and reduce.

Example 1: Power Rule

Find \(\dfrac{d}{dx}\!\left[5x^3 - 2\sqrt{x} + \dfrac{4}{x}\right]\)

1
Rewrite: \(5x^3 - 2x^{1/2} + 4x^{-1}\)
2
Operation: This is a sum/difference of power terms. Differentiate each term.
3
Apply power rule term by term:
\(\frac{d}{dx}[5x^3] = 15x^2\)
\(\frac{d}{dx}[-2x^{1/2}] = -x^{-1/2}\)
\(\frac{d}{dx}[4x^{-1}] = -4x^{-2}\)
4
Result: \(\boxed{15x^2 - \dfrac{1}{\sqrt{x}} - \dfrac{4}{x^2}}\)

Example 2: Product Rule

Find \(\dfrac{d}{dx}\!\left[x^2 \sin x\right]\)

1
Identify: Two functions multiplied: \(f(x) = x^2\) and \(g(x) = \sin x\).
2
Differentiate each: \(f'(x) = 2x\), \(g'(x) = \cos x\).
3
Apply product rule: \(f'g + fg' = 2x \sin x + x^2 \cos x\)

Example 3: Chain Rule

Find \(\dfrac{d}{dx}\!\left[\sin(3x^2)\right]\)

1
Identify composition: Outer: \(f(u) = \sin u\). Inner: \(g(x) = 3x^2\).
2
Differentiate outer: \(f'(u) = \cos u\), evaluated at \(u = 3x^2\) gives \(\cos(3x^2)\).
3
Differentiate inner: \(g'(x) = 6x\).
4
Multiply: \(f'(g(x)) \cdot g'(x) = \cos(3x^2) \cdot 6x\)

Example 4: Combined Rules

Find \(\dfrac{d}{dx}\!\left[e^x \cdot \cos(2x)\right]\)

1
Outermost operation: Product of \(f(x) = e^x\) and \(g(x) = \cos(2x)\). Use the product rule.
2
Differentiate \(f\): \(f'(x) = e^x\).
3
Differentiate \(g\): \(\cos(2x)\) is a composition. By the chain rule: \(g'(x) = -\sin(2x) \cdot 2 = -2\sin(2x)\).
4
Combine: \(f'g + fg' = e^x\cos(2x) + e^x \cdot (-2\sin(2x))\)

Common Mistakes

✕ Forgetting the Chain Rule
Writing \(\frac{d}{dx}[\sin(3x)] = \cos(3x)\) instead of \(3\cos(3x)\). Always multiply by the inner derivative.
✕ Product Rule Sign Error
Using subtraction in the product rule. Remember: it is \(f'g + fg'\) with a plus sign, not minus.
✕ Quotient Rule Order
Swapping \(f'g - fg'\) to \(fg' - f'g\). The numerator derivative comes first: "lo d-hi minus hi d-lo."

Rule Selection Summary

You See Rule Formula
\(cx^n\), sums of powers Power Rule \(nx^{n-1}\)
\(f(x) \cdot g(x)\) Product Rule \(f'g + fg'\)
\(\dfrac{f(x)}{g(x)}\) Quotient Rule \(\dfrac{f'g - fg'}{g^2}\)
\(f(g(x))\) Chain Rule \(f'(g(x)) \cdot g'(x)\)
Multiple layers Combine rules Outer first, then inner

Question 1 of 5

Which differentiation rule should you use first to find \(\dfrac{d}{dx}[x^3 \cdot e^x]\)?

Question 2 of 5

What is \(\dfrac{d}{dx}[\cos(5x)]\)?

Question 3 of 5

What is the derivative of \(\dfrac{x^2 + 1}{x}\)?

Question 4 of 5

To differentiate \(f(x) = (2x+1)^5 \cdot \ln x\), which combination of rules is needed?

Question 5 of 5

What is \(\dfrac{d}{dx}\!\left[e^{x^2}\right]\)?