Stewart Calculus · 8th Edition
Chapter 3: Differentiation Rules
Every key formula from Stewart Calculus Chapter 3, in one searchable page. Click a card to study it — worked examples included.
3.1 · POWER & EXPONENTIAL RULES
The basics — power rule works for any real n
d/dx (c) = 0 · d/dx (xn) = nxn−1
(cf)′ = cf′ · (f ± g)′ = f′ ± g′
(cf)′ = cf′ · (f ± g)′ = f′ ± g′
Exponentials
d/dx (ex) = ex · d/dx (bx) = bx ln b
KEY NOTES
- Rewrite first, then differentiate: x½ and x−1 are power-rule ready.
3.2 · PRODUCT & QUOTIENT RULES
(fg)′ = f′g + fg′
(f/g)′ = f′g − fg′g²
(f/g)′ = f′g − fg′g²
KEY NOTES
- (fg)′ ≠ f′g′ — the derivative of a product is NOT the product of derivatives.
WORKED EXAMPLE
f(x) = x² + 3xx − 1 ⇒ f′ = (2x+3)(x−1) − (x²+3x)(x − 1)²
= x² − 2x − 3(x − 1)².
= x² − 2x − 3(x − 1)².
3.3 · TRIG DERIVATIVES
(sin x)′ = cos x · (cos x)′ = −sin x
(tan x)′ = sec²x · (cot x)′ = −csc²x
(sec x)′ = sec x tan x · (csc x)′ = −csc x cot x
(tan x)′ = sec²x · (cot x)′ = −csc²x
(sec x)′ = sec x tan x · (csc x)′ = −csc x cot x
All proved from limθ→0 sin θ/θ = 1.
KEY NOTES
- Cofunctions take a minus: cos → −sin, cot → −csc², csc → −csc cot.
3.4–3.5 · CHAIN RULE& IMPLICIT DIFFERENTIATION
Chain rule
dy/dx = dydu · dudx
Outside first, then inside: d/dx sin(x²) = cos(x²) · 2x.
WORKED EXAMPLE
f(x) = = (x²+1)½ ⇒
f′ = ½(x²+1)−½ · 2x = x.
f′ = ½(x²+1)−½ · 2x = x.
Implicit differentiation
Differentiate both sides w.r.t. x; y-terms pick up a y′ factor (chain rule) — then solve for y′.
WORKED EXAMPLE
x² + y² = 25 ⇒ 2x + 2y · y′ = 0
⇒ y′ = −x/y (at (3, 4): −¾).
⇒ y′ = −x/y (at (3, 4): −¾).
3.6 · LOG DERIVATIVES
d/dx (ln x) = 1x · d/dx (logb x) = 1x ln b
d/dx ln(g(x)) = g′(x)g(x)
d/dx ln(g(x)) = g′(x)g(x)
Logarithmic differentiation — for xx-type functions
y = xx ⇒ ln y = x ln x ⇒ y′y = ln x + 1 ⇒ y′ = xx(ln x + 1)
KEY NOTES
- The power rule does NOT apply to xx — the base must be constant.
3.7–3.11 · APPLICATIONS
Related rates
Differentiate the relating equation w.r.t. t (chain rule!) — substitute known values LAST.
WORKED EXAMPLE
Ripple: A = πr², dr/dt = 2 cm/s ⇒ dA/dt = 2πr · dr/dt.
At r = 5: dA/dt = 20π cm²/s.
At r = 5: dA/dt = 20π cm²/s.
Linear approximation & differentials
L(x) = f(a) + f′(a)(x − a) · dy = f′(x) dx
Growth / decay & hyperbolic
y′ = ky ⇒ y = Cekt (doubling / halving time) · sinh x = (ex−e−x)/2, cosh x = (ex+e−x)/2; (sinh x)′ = cosh x, (cosh x)′ = sinh x.
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