3.5 Selecting Procedures for Calculating Derivatives

Match the rule to the structure

Before differentiating, identify the function's structure: product (f·g → product rule), quotient (f/g → quotient rule), composition (f(g(x)) → chain rule), or implicit relation (x and y mixed → implicit differentiation). Many problems require combining multiple rules, so scan the expression outside-in to plan your approach before writing derivatives.

Layering rules together

Complex expressions often nest rules: a quotient of products, or a chain rule wrapped around a product. Work from the outermost operation inward — apply chain rule first to composite outer functions, then handle inner products/quotients. Keep track of parentheses and don't forget the derivative of the inner function when using the chain rule.

Choosing efficient strategies

Sometimes rewriting a function first simplifies differentiation: use log properties before differentiating logarithms, rewrite radicals as rational exponents, or use implicit differentiation instead of solving explicitly for y. Recognizing when a shortcut (like logarithmic differentiation for products/quotients/powers) saves algebra is a key exam skill.

Key terms

How to recognize when to use the chain rule

When the function is a composition — one function plugged into another, like f(g(x)).

How to recognize when to use the product rule

When two separate functions of x are multiplied together, like f(x)·g(x).

How to recognize when to use the quotient rule

When one function of x is divided by another, like f(x)/g(x).

When to use implicit differentiation

When x and y are mixed in an equation and y can't be easily isolated as a function of x.

Strategy tip for complex derivatives

Simplify first when possible (log properties, rewriting radicals as exponents) and work from the outermost structure inward, combining rules as needed.

Try an AP-style question

d/dx (x² e^(3x)) = ?

1. 2x e^(3x) + 3x² e^(3x)
2. 2x e^(3x)
3. 2x e^(3x) + x² e^(3x)
4. 6x e^(3x)