Module 3 Quick Reference

Differentiation Rules – Calculus I

Product Rule

(fg)' = f'g + fg'

Extended: (fgh)' = f'gh + fg'h + fgh'

Quotient Rule

(f/g)' = (f'g − fg') / g²

Tip: "low d-high minus high d-low, over low squared"

Chain Rule

d/dx [f(g(x))] = f'(g(x)) · g'(x)
General Power: d/dx [un] = n un−1 · u'

Trig Derivatives

d/dx [sin x] = cos x

d/dx [tan x] = sec² x

d/dx [sec x] = sec x tan x

d/dx [cos x] = −sin x

d/dx [cot x] = −csc² x

d/dx [csc x] = −csc x cot x

Common Chain Rule Patterns

d/dx [eu] = eu · u'

d/dx [ln u] = u'/u

d/dx [sin u] = cos u · u'

d/dx [cos u] = −sin u · u'

Implicit Differentiation

1. Differentiate both sides w.r.t. x

2. Each y-term gets × dy/dx (Chain Rule)

3. Collect dy/dx terms, factor, solve

x² + y² = r² ⇒ dy/dx = −x/y

Strategy Checklist