AP Calculus AB/BC · Unit 3 · Topic 3.5
Selecting Procedures for Calculating Derivatives
Decompose mixed expressions and choose product, quotient, chain, implicit, or inverse rules in a workable order.
1. Topic Focus
Differentiate nested, implicit, inverse, and higher-order relationships by choosing procedures that match the function structure.
This topic: Decompose mixed expressions and choose product, quotient, chain, implicit, or inverse rules in a workable order.
2. Key Relationship
Read every symbol with its domain, direction, units, and hypotheses before applying the relationship.
3. Visual Connection
4. Worked Example
For e^(x²)sin x, use product and chain rules together.
Write the governing relationship first, carry out the algebra cleanly, and finish with a sentence that answers the mathematical question.
5. Concept Development
Selecting a differentiation procedure begins with the structure of the expression, not with whichever formula first comes to mind. Read the expression as a tree: identify its outermost operation, apply the rule for that operation, and then move inward to differentiate each component.
| Top-level structure | First procedure | Question to ask next |
|---|---|---|
| \(f(x)+g(x)\) | Sum or difference rule | Which rule differentiates each term? |
| \(f(x)g(x)\) | Product rule | Is either factor itself composite? |
| \(f(x)/g(x)\) | Quotient rule or valid rewrite | Can the expression be simplified without changing its domain? |
| \(f(g(x))\) | Chain rule | How many nested layers are present? |
| Equation mixing \(x\) and \(y\) | Implicit differentiation | Where must product and chain rules introduce \(y'\)? |
| \(f^{-1}(x)\) | Inverse-function rule | Which original input maps to the target output? |
Top-level punctuation matters. In \(x^2e^{\sin x}\), the multiplication is outside every other operation, so the product rule comes first. In \(e^{x^2\sin x}\), the exponential is outside, so the chain rule comes first and the exponent then requires the product rule. Similar symbols can therefore demand different procedure orders.
Core rules available for the plan.
These rules can appear at different levels of the same expression. State the outer rule first, preserve the undifferentiated factors or inner inputs it requires, and then differentiate the nested pieces.
A five-step planning routine.
- Mark top-level sums and differences; they divide the expression into independent terms.
- For each term, identify whether the outside is a product, quotient, composition, inverse, or implicit relation.
- Write a skeleton of the selected rule before substituting complicated pieces.
- Repeat the structural analysis inside every factor or input.
- Simplify only after the derivative is structurally complete, then check domain and signs.
When simplifying first helps. Expanding a small polynomial product, rewriting a quotient as a negative power, or canceling a common factor can shorten the derivative. However, an algebraically simplified expression may have a larger domain than the original. For example,
The simplified rule has derivative 1 on the original domain, but the original function still has no derivative at \(x=1\) because it is not defined there.
Symbolic formula or value only? If the problem asks for a derivative at one point and supplies tables or graphs, build the rule symbolically and substitute only the required values. Do not try to invent formulas for the tabulated functions. If the relation is implicit, it may also be faster to differentiate first and then substitute the point before fully solving for \(y'\).
Verification. A reliable derivative has compatible domain restrictions, expected units, plausible signs, and the right number of product-rule terms and chain-rule factors. When practical, differentiate an equivalent form by a second method. Agreement is strong evidence; disagreement identifies where the procedure broke down.
6. Detailed Worked Example and Error Check
Example 1: Product outside, chain inside. For \(F(x)=e^{x^2}\sin x\), the top-level operation is multiplication:
The product rule creates two terms; the factor \(2x\) appears only when differentiating the composite exponential.
Example 2: Chain outside, product inside. For \(G(x)=e^{x^2\sin x}\), begin with the exponential chain:
This derivative has one top-level term, unlike Example 1, because the multiplication lies inside the exponent.
Example 3: Quotient with a composite numerator. Let
The quotient rule comes first; its numerator derivative needs the chain rule:
The original and derivative formulas exclude \(x=k\pi\), where the denominator is zero.
Example 4: Simplify while preserving the original domain.
Thus \(P'(x)=1\) for \(x\ne1\). The cancellation removes a factor from the formula, not the hole from the original function.
Example 5: Implicit relation with several rules. For
select implicit differentiation. The right side requires an exponential chain and the product rule inside its exponent:
Collecting the \(y'\)-terms gives
where the denominator is nonzero.
Example 6: Inverse function and an inner function. Let \(Q(x)=f^{-1}(x^2+3)\). Suppose \(f(2)=7\) and \(f'(2)=-3\). At \(x=2\), the inner value is 7 and \(f^{-1}(7)=2\), so
Example 7: Use table values without finding formulas. Suppose \(R(x)=f(g(x))/x\) and
The top-level quotient contains a composition:
AP error check. Do not treat a product as a composition, multiply derivatives across a product, divide derivatives across a quotient, omit an inner derivative, substitute table values at the wrong input, cancel across addition, or let simplification silently change the reported domain.
7. AP Reasoning Routine
Mark inner and outer functions, track every derivative factor, solve algebraically for the requested derivative, and verify the result's domain.
- Identify the representation and requested quantity.
- State the rule or theorem and verify its conditions.
- Keep exact values until the final requested approximation.
- Interpret sign, units, interval, and context.
For each expression, name the top-level procedure before calculating.
(a) Differentiate \(y=e^{x^2}\cos x\).
(b) Differentiate \(y=(x^2+1)^4/(x-2)\).
(c) Find \(dy/dx\) if \(x^2y+\sin y=4\).
(d) Let \(H(x)=f^{-1}(3x-1)\). If \(f(4)=5\) and \(f'(4)=6\), find \(H'(2)\).
(e) Simplify first and differentiate \(P(x)=(x^3-8)/(x-2)\), stating the derivative domain.
(f) Let \(R(x)=f(g(x))/x\), with \(g(1)=2\), \(g'(1)=-3\), \(f(2)=5\), and \(f'(2)=4\). Find \(R'(1)\).
(g) Explain why \(x^2\sin x\) and \(\sin(x^2)\) require different top-level rules.
(h) Differentiate \(y=\arcsin(x^2)+\ln(3x+1)\) and state the derivative domain.
Check the solution
(a) The top-level rule is the product rule: \(y'=2xe^{x^2}\cos x-e^{x^2}\sin x=e^{x^2}(2x\cos x-\sin x)\).
(b) Use the quotient rule, then the chain rule: \(y'=[8x(x^2+1)^3(x-2)-(x^2+1)^4]/(x-2)^2\).
(c) Use implicit differentiation. From \(2xy+x^2y'+\cos y\,y'=0\), obtain \(y'=-2xy/(x^2+\cos y)\), where the denominator is nonzero.
(d) The outer rule is the inverse-function rule with a chain factor: \(H'(2)=3/f'(f^{-1}(5))=3/f'(4)=1/2\).
(e) For \(x\ne2\), \(P(x)=x^2+2x+4\), so \(P'(x)=2x+2\). The derivative domain remains \(x\ne2\).
(f) The quotient rule and chain rule give \(R'(1)=[1\cdot4(-3)-5]/1^2=-17\).
(g) In \(x^2\sin x\), multiplication is top-level, so use the product rule. In \(\sin(x^2)\), sine is outside a composition, so use the chain rule.
(h) Split the top-level sum, then use chain rules: \(y'=2x/\sqrt{1-x^4}+3/(3x+1)\). The derivative requires \(|x|<1\) and \(x>-1/3\), so its real domain is \((-1/3,1)\).