Short on time? This page is the recap. For explanations, the structure tree and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: f′(x) and dy/dx are derivatives; d/dx [ … ] means “the derivative of”. Angles are in radians.
Recap
- Topic 3.5 has no new rule. It tests whether you pick the right one.
- Step 1: rewrite if it helps. Roots and reciprocals to powers; split a fraction with a single-term bottom; use log properties.
- Step 2: find the last operation (the one you would do last if you evaluated by hand). It picks the first rule.
- Step 3: repeat on each piece until every piece is a basic function.
- With table values, write the rule with letters first, then substitute.
Key relationships
| Form | Rule |
|---|---|
| xⁿ | n xⁿ⁻¹ |
| k·f, f ± g | k·f′, f′ ± g′ |
| f·g | f′g + fg′ |
| f/g | (f′g − fg′)/g² |
| f(g(x)) | f′(g(x))·g′(x) |
| sin x, cos x, tan x | cos x, −sin x, sec²x |
| sec x, csc x, cot x | sec x tan x, −csc x cot x, −csc²x |
| eˣ, ln x | eˣ, 1/x |
| arcsin x, arccos x, arctan x | 1/√(1 − x²), −1/√(1 − x²), 1/(1 + x²) |
| x and y mixed in an equation | implicit: differentiate both sides, solve for dy/dx |
| f⁻¹ at b, where f(a) = b | (f⁻¹)′(b) = 1/f′(a), needs f′(a) ≠ 0 |
Assumptions behind the rules
- The functions involved are differentiable at the point you use.
- Trig rules need radians.
- Log rewrites such as ln(x⁵) = 5 ln x hold only where the logs are defined.
- The inverse-function rule needs f one-to-one near a and f′(a) ≠ 0.
Mistakes to avoid
- (fg)′ = f′g′ or (f/g)′ = f′/g′. Both are wrong.
- Forgetting the inner derivative in a chain rule step.
- f′(1) instead of f′(g(1)) when differentiating f(g(x)) at x = 1.
- 1/f′(b) instead of 1/f′(a) for an inverse.
- Quotient rule with a single-term bottom, then an algebra slip. Rewrite as powers instead.
- Sign errors in the quotient rule: the order is f′g − fg′.
Quick self-check
- Differentiate 4/√x. (−2x^(−3/2): rewrite as 4x^(−1/2) first)
- Find the derivative of (x² + 1)⁵ at x = 1. (160: chain rule gives 5(x² + 1)⁴ · 2x = 5 × 16 × 2)
- Differentiate eˣ sin x. (eˣ(sin x + cos x): product rule)
- Which rule comes first for sin(x² eˣ)? (The chain rule, since sin is applied last. The inside then needs the product rule.)
Next: practice questions.