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Definition of the derivative
f′(x) = lim h→0 [f(x + h) − f(x)] ÷ h
lim x→0 sin(x)/x
1
lim x→0 (1 − cos x)/x
0
lim x→∞ (1 + 1/x)ˣ
e
L'Hôpital's rule
If a limit gives 0/0 or ∞/∞, then lim f/g = lim f′/g′
Continuity at x = a
f(a) exists, the limit exists, and the limit equals f(a)
Power rule
d/dx xⁿ = n·xⁿ⁻¹
Product rule
(uv)′ = u′v + uv′
Quotient rule
(u/v)′ = (u′v − uv′) ÷ v²
Chain rule
d/dx f(g(x)) = f′(g(x))·g′(x)
d/dx sin(x)
cos(x)
d/dx cos(x)
−sin(x)
d/dx tan(x)
sec²(x)
d/dx eˣ
eˣ
d/dx ln(x)
1/x
d/dx aˣ
aˣ·ln(a)
d/dx arctan(x)
1/(1 + x²)
∫ xⁿ dx (n ≠ −1)
xⁿ⁺¹/(n + 1) + C
∫ 1/x dx
ln|x| + C
∫ eˣ dx
eˣ + C
∫ sin(x) dx
−cos(x) + C
∫ cos(x) dx
sin(x) + C
Fundamental Theorem of Calculus
∫ from a to b of f(x) dx = F(b) − F(a), where F′ = f
u-substitution
∫ f(g(x))·g′(x) dx = ∫ f(u) du with u = g(x)
Critical point
Where f′(x) = 0 or f′(x) is undefined
Second derivative test
f′(c) = 0 and f″(c) > 0 → local min; f″(c) < 0 → local max
Average value of f on [a, b]
1/(b − a) · ∫ from a to b of f(x) dx
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