Unit 3.5 – Selecting Procedures for Calculating Derivatives

AP® Calculus AB & BC | How to Pick the Right Differentiation Rule Every Time

Keen differentiation skills are essential for the AP® Calculus exam. This section guides you through the art of selecting the most efficient (and required!) differentiation method for any given problem, whether it’s explicit or implicit, composite or inverse.

🚦 Step-by-Step Procedure Selection

Ask these questions (in order) when you approach a derivative:

Can you simplify or rewrite? (Algebra first: Expand, rewrite as powers, simplify roots, etc.)
Is it a sum, difference, or constant multiple?
Apply the linear rules: break apart and differentiate term-by-term.
Is it a simple power of x or basic function?
Use the Power Rule, constant, exponential, log/trig rules as needed.
Is it multiplication (product)?
Use the Product Rule: $$ (uv)' = u'v + uv' $$
Is it a division (quotient)?
Use the Quotient Rule: $$ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} $$
Is it a composite (function inside function)?
Use the Chain Rule: $$ (f(g(x)))' = f'(g(x)) \cdot g'(x) $$
Does it involve x and y (implicit)?
Use Implicit Differentiation: Differentiate both sides, apply $$\frac{dy}{dx}$$ as needed for y terms.
Is it an inverse function?
Use the Inverse Rule: $$ (f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))} $$

If you recognize multiple rules in a single expression, apply them in the correct order (often outside-in).

🛠️ Quick Reference Table

Problem Type	What To Look For	Rule(s) To Use
Simple power/monomial	$$x^n$$, constants	Power rule, constant rule
Sum, difference	"+" or "-" between terms	Linearity: D(f \pm g) = f' \pm g'
Product	u(x) \* v(x)	Product rule
Quotient	u(x)/v(x)	Quotient rule
Composition	f(g(x)), nested functions	Chain rule
Implicit	Both x and y in one equation	Implicit differentiation
Inverse	$$y = f^{-1}(x)$$, or table with f,f'	Inverse function rule

💡 Short Notes, Tricks & Common Patterns

Product inside chain? Chain rule takes priority, then product if you have both.
Roots & reciprocals: Rewrite as powers before starting to use the power or chain rule.
For complicated polynomials: Expand first!
Trig, log, exp: Know base derivative rules and always check if chain is needed (e.g. $$\sin(3x)$$).
Multiple rules: Work from outermost (composition) to inner (product/quotient), unless simpler to fully expand.
Table of values for f and f': For inverse problems, match x and y carefully and use the inverse formula.

📖 Worked Examples

Example 1: $$f(x) = (3x^2 - 4x)^5$$

Composite: Chain Rule
$$5(3x^2-4x)^4 \cdot (6x-4)$$

Example 2: $$g(x) = x^2\sin(2x)$$

Product Rule + Chain Rule
$$2x\sin(2x) + x^2\cos(2x)\cdot 2 = 2x\sin(2x) + 2x^2\cos(2x)$$

Example 3: $$h(x) = \frac{e^x}{x^2+1}$$

Quotient Rule
$$\frac{e^x(x^2+1) - e^x\cdot2x}{(x^2+1)^2}$$

Example 4 (implicit): $$x^2+y^2=10$$

Implicit Diff:
$$2x+2y\frac{dy}{dx}=0 \rightarrow \frac{dy}{dx}=-\frac{x}{y}$$

Example 5 (inverse): Table – If $$f(1)=3, f'(1)=7$$, find $$(f^{-1})'(3)$$

$$(f^{-1})'(3)=\frac{1}{f'(1)}=\frac{1}{7}$$

📝 Practice Problems

Try These Yourself:

$$\frac{d}{dx}[ (x^4+2x)^7 ]$$
$$\frac{d}{dx}[ x^2 e^{3x} ]$$
$$\frac{d}{dx}[ \tan x / x ]$$
$$\frac{d}{dx}[ \ln(x^2+5) ]$$
$$\frac{d}{dx}$$ if $$ y^3 + xy = 7 $$

Answers:

$$7(x^4+2x)^6 (4x^3+2)$$
$$2x e^{3x} + x^2\cdot 3e^{3x} = 2x e^{3x} + 3x^2 e^{3x}$$
$$\frac{x\sec^2 x - \tan x}{x^2}$$
$$\frac{2x}{x^2+5}$$
$$\frac{dy}{dx}=\frac{-y}{3y^2+x}$$

✏️ AP® Exam Tips – Derivative Selection

Explicitly name your rule: "By the Chain/Product/Quotient Rule…" for FRQs
For multiple rules, clearly show each step and what’s being used
Check for hidden compositions—most mistakes are missing chain rule!
Expand before differentiating if it simplifies the process
Take your time identifying structure, then work methodically
Box your final answer and clearly label terms on AP® free response