Transcendental Functions

ln 1 = 0  ·  ln e = 1
ln(xy) = ln x + ln y
ln(x/y) = ln x − ln y
ln(xⁿ) = n ln x
d/dx [ln x] = 1/x

e = 2.71828…
eˣ = inverse of ln x
d/dx [eˣ] = eˣ
e^(ln x) = x  ·  ln(eˣ) = x
d/dx [aˣ] = aˣ · ln a

d/dx [ln(x²+1)] = 2x/(x²+1)
d/dx [e^(3x)] = 3e^(3x)
d/dx [x^x] = x^x(1+ln x)
d/dx [ln|cos x|] = −tan x

y = x^x   →   ln y = x ln x
(1/y)y' = ln x + 1
y' = x^x(ln x + 1)

y₀e^(kT) = y₀/2
e^(kT) = 1/2   →   kT = −ln 2
T = ln 2 / |k|
Carbon-14: T ≈ 5730 ปี

y₀e^(kT) = 2y₀
e^(kT) = 2   →   kT = ln 2
T = ln 2 / k
กฎ Rule of 70: T ≈ 70/r% ปี

dT/dt = k(T − Tₛ)
T(t) = Tₛ + (T₀−Tₛ)e^(kt)
k < 0: วัตถุเย็นลงสู่อุณหภูมิห้อง Tₛ

A = P·e^(rt) (continuous)
r = อัตราดอกเบี้ย, t = ปี
P=1000, r=5%, t=10:
A = 1000·e^(0.5) ≈ 1648.72

∫ (1/x) dx = ln|x| + C
∫ eˣ dx = eˣ + C
∫ aˣ dx = aˣ/ln(a) + C
∫ ln x dx = x ln x − x + C

∫ 1/√(1−x²) dx = arcsin x + C
∫ 1/(1+x²) dx = arctan x + C
∫ 1/(x√(x²−1)) dx = arcsec|x| + C
∫ 1/√(a²−x²) dx = arcsin(x/a) + C

∫ tan x dx = ln|sec x| + C
∫ sec x dx = ln|sec x + tan x| + C
∫ cot x dx = ln|sin x| + C
∫ csc x dx = −ln|csc x + cot x| + C

∫ x/(x²+1) dx = (1/2)ln(x²+1) + C
∫ e^(2x) dx = (1/2)e^(2x) + C
∫₀¹ 1/(1+x²) dx = arctan(1) − 0 = π/4
∫₁ᵉ (1/x) dx = ln e − ln 1 = 1

cosh²x − sinh²x = 1
1 − tanh²x = sech²x
sinh(x+y) = sinh x cosh y + cosh x sinh y
cosh(x+y) = cosh x cosh y + sinh x sinh y

d/dx[sinh x] = cosh x
d/dx[cosh x] = sinh x
d/dx[tanh x] = sech²x
d/dx[sech x] = −sech x · tanh x

∫ sinh x dx = cosh x + C
∫ cosh x dx = sinh x + C
∫ sech²x dx = tanh x + C
∫ 1/√(x²+1) dx = sinh⁻¹x + C

รูปร่างของสายโซ่ที่แขวน:
y = a·cosh(x/a)
Gateway Arch ใช้ inverted catenary
y = a·cosh(x/b) เป็นรูปโค้ง