3.5 Selecting Procedures for Calculating Derivatives

Learning objectives

3.5Selecting Procedures for Calculating Derivatives—Topic 3.5• 3.5 This topic is intended to focus on the skill of selecting an appropriate procedure for calculating derivatives. Students should be given opportunities to practice when and how to apply all learning objectives relating to calculating derivatives.

Choose a derivative procedure from the function's structure before doing algebra. First identify how the largest pieces are connected; then inspect each piece for additional rules.

Sum or difference of terms → differentiate term by term.\n• Product of two changing factors → product rule.\n• Quotient of two changing expressions → quotient rule, unless rewriting makes a simpler equivalent form.\n• One function inside another → chain rule.\n• Equation linking $x$ and $y$ → implicit differentiation.\n• Derivative of an inverse at a corresponding point → inverse-function derivative rule.

Simplify only when it makes the structure clearer.\n2. Mark the outermost operation.\n3. Select the rule for that operation.\n4. Work inward and attach any additional rules, especially the chain rule for composite factors.\n5. Check that every nonconstant factor has been differentiated where the selected rule requires it.

For $y=x^2\sin(3x)$ , the outer structure is a product, so use the product rule. The second factor is composite, so its derivative also needs the chain rule: $\frac{dy}{dx}=2x\sin(3x)+x^2\bigl(3\cos(3x)\bigr).$ Product rule organizes the two factors; chain rule differentiates the inner input $3x$ .

Do not choose a rule from a symbol in isolation. Parentheses may indicate multiplication or composition depending on context. Also, $(uv)'$ is not $u'v'$ and $(f(g(x)))'$ is not just $f'(g(x))$ : the first needs two product-rule terms, while the second needs the inner derivative $g'(x)$ .

Next Topic

—3.6 Calculating Higher-Order DerivativesRead