AP Calculus AB / Unit 3: Differentiation: Composite, Implicit, and Inverse Functions
3.5 Selecting Procedures for Calculating Derivatives
Notes
Look at the structure first:
- A product → product rule. A quotient → quotient rule (or rewrite as a product).
- A function inside another → chain rule.
- An equation not solved for $y$ → implicit differentiation.
- Simplify first when you can; it saves work.
Practice
Best first step for $\displaystyle \frac{d}{dx}\frac{x^3+x}{x}$?
Simplifying gives $2x$ directly.
$\displaystyle \frac{d}{dx}[x\cos(3x)]$ needs:
$\cos(3x)$ needs the chain rule inside the product.
$\displaystyle \frac{d}{dx}[x\cos(3x)]=$
$1\cdot\cos3x+x(-3\sin3x)$.
Ready for more? Take the Unit 3 test.