Short on time? This page is the recap. For explanations, the sine cycle diagram and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- Second derivative: differentiate f′ to get f″. Keep going for higher orders.
- Existence: f″(a) exists only if f′ is differentiable at a. Example: x^(4/3) has f′(0) = 0 but no f″(0).
- Simplify f′ first, then classify it and choose the next rule (Topic 3.5). The rule for f′ is often different from the rule for f: tan x needs a known result, but sec²x then needs the chain rule.
- Implicit curves: differentiate dy/dx again, remembering y depends on x. Substitute dy/dx, then use the curve’s equation to simplify.
Key relationships
| Idea | Fact |
|---|---|
| Second derivative notations | f″(x), y″, d²y/dx² |
| nth derivative notations | f⁽ⁿ⁾(x), y⁽ⁿ⁾, dⁿy/dxⁿ |
| d²y/dx² | means d/dx (dy/dx), not (dy/dx)² |
| Polynomial of degree n | nth derivative is constant; all later ones are 0 |
| e^(kx) | nth derivative is kⁿ e^(kx) |
| sin x | cycle: cos x, −sin x, −cos x, sin x (period 4) |
| sin(kx) or cos(kx) | each step adds a factor k; use n ÷ 4 remainder for the function |
| Implicit | derivative of y is y′; derivative of y′ is y″ |
Assumptions behind the method
- Each derivative you take must exist on the interval you use.
- Trig patterns need radians.
- In an implicit second derivative, the point must lie on the curve, and the bottom of dy/dx must not be 0 there.
Mistakes to avoid
- Squaring dy/dx instead of differentiating it again.
- Reading f⁽⁴⁾(x) as [f(x)]⁴.
- Treating y as a constant in an implicit second derivative.
- Dropping the chain factor k at each step for sin(kx), cos(kx) or e^(kx).
- Sign slips in the sine cycle. Write out all four steps once.
- Assuming f″ exists because f′ does.
Quick self-check
- y = x⁶. Find y‴. (120x³)
- Find d²/dx² [e⁵ˣ]. (25e⁵ˣ)
- Let f(x) = x sin x. Find f″(x). (2 cos x − x sin x: product rule twice)
- What is the 10th derivative of x⁹ + 4x? (0: a degree-9 polynomial has a constant 9th derivative)
Next: practice questions.