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Transcendental Functions Cheat Sheet

Complete reference for exponential, logarithmic, trigonometric, and hyperbolic functions with derivatives, integrals, and identities.

📚 What are Transcendental Functions?

Functions that "transcend" algebra—cannot be expressed as finite combinations of algebraic operations (addition, multiplication, roots).

Exponential

eˣ, aˣ, e^(f(x))

Logarithmic

ln(x), logₐ(x), ln(f(x))

Trigonometric

sin(x), cos(x), tan(x), etc.

Hyperbolic

sinh(x), cosh(x), tanh(x), etc.

🚀 Exponential Functions
Function Derivative Integral
eˣ eˣ eˣ + C
aˣ (a>0, a≠1) aˣ·ln(a) aˣ/ln(a) + C
e^(kx) k·e^(kx) e^(kx)/k + C
x·eˣ eˣ(x+1) eˣ(x-1) + C
💡 Key Property: eˣ is its own derivative and integral! This makes it unique among all functions.
Example: d/dx [3·e^(2x)] = 3·2·e^(2x) = 6e^(2x)
Graph Behavior: eˣ grows faster than any polynomial. As x→∞, eˣ→∞. As x→-∞, eˣ→0.
📊 Logarithmic Functions
Function Derivative Integral
ln|x| 1/x x·ln|x| - x + C
logₐ|x| 1/(x·ln(a)) (x·ln|x| - x)/ln(a) + C
ln|f(x)| f'(x)/f(x) Special techniques
x·ln(x) ln(x) + 1 (x²/2)(ln(x)-½) + C
💡 Change of Base: logₐ(x) = ln(x)/ln(a). Use this to convert to natural log.
Example: d/dx [ln(x²+1)] = (2x)/(x²+1)
Graph Behavior: ln(x) defined for x>0. As x→0⁺, ln(x)→-∞. As x→∞, ln(x)→∞ (slowly).
🔄 Trigonometric Functions
Function Derivative Integral
sin(x) cos(x) -cos(x) + C
cos(x) -sin(x) sin(x) + C
tan(x) sec²(x) ln|sec(x)| + C
cot(x) -csc²(x) ln|sin(x)| + C
sec(x) sec(x)tan(x) ln|sec(x)+tan(x)| + C
csc(x) -csc(x)cot(x) -ln|csc(x)+cot(x)| + C
💡 Memory Aid: Derivatives of co-functions have negatives. sin↔cos, tan↔sec, cot↔csc follow patterns.
Example: ∫ tan(x) dx = ∫ sin(x)/cos(x) dx = -ln|cos(x)| + C = ln|sec(x)| + C
🎯 Inverse Trigonometric
Function Derivative Domain
sin⁻¹(x) 1/√(1-x²) [-1, 1]
cos⁻¹(x) -1/√(1-x²) [-1, 1]
tan⁻¹(x) 1/(1+x²) (-∞, ∞)
cot⁻¹(x) -1/(1+x²) (-∞, ∞)
sec⁻¹(x) 1/(|x|√(x²-1)) |x| ≥ 1
csc⁻¹(x) -1/(|x|√(x²-1)) |x| ≥ 1
💡 Pattern: Derivatives of inverse trig functions often involve 1/(1±x²) or 1/√(1±x²).
Example: d/dx [tan⁻¹(3x)] = 3/(1+(3x)²) = 3/(1+9x²)
🔥 Hyperbolic Functions
Function Derivative Identity
sinh(x) = (eˣ-e⁻ˣ)/2 cosh(x) cosh²x - sinh²x = 1
cosh(x) = (eˣ+e⁻ˣ)/2 sinh(x) 1 - tanh²x = sech²x
tanh(x) = sinh(x)/cosh(x) sech²(x) coth²x - 1 = csch²x
sech(x) = 1/cosh(x) -sech(x)tanh(x) sinh(2x) = 2sinh(x)cosh(x)
💡 Connection to Trig: Similar to trig but without "i": cosh²x - sinh²x = 1 (compare to cos²x + sin²x = 1)
Example: d/dx [cosh(2x)] = 2·sinh(2x)
🔗 Essential Identities
Type Identity Use
Exponential eˣ·eʸ = eˣ⁺ʸ Simplify products
Logarithmic ln(xy) = ln(x) + ln(y) Expand logs
Trig sin²x + cos²x = 1 Pythagorean identity
Angle Sum sin(x+y) = sinx·cosy + cosx·siny Expand trig functions
Double Angle sin(2x) = 2sinx·cosx Simplify expressions
Euler's e^(ix) = cos(x) + i·sin(x) Connects exponential and trig
💡 Euler's Formula: The most beautiful equation in mathematics: e^(iπ) + 1 = 0

📋 Common Composite Function Derivatives

Function Derivative Chain Rule Application
e^(f(x)) f'(x)·e^(f(x)) Multiply by derivative of exponent
ln|f(x)| f'(x)/f(x) Derivative over original function
sin(f(x)) f'(x)·cos(f(x)) Derivative times cos of inside
cos(f(x)) -f'(x)·sin(f(x)) Negative derivative times sin of inside
a^(f(x)) f'(x)·a^(f(x))·ln(a) Derivative times original times ln(a)

∫ Transcendental Integration Techniques

By Parts for Transcendentals

∫ eˣ·sin(x) dx = (eˣ/2)(sin x - cos x) + C

Trig Substitution

∫ 1/√(a²-x²) dx = sin⁻¹(x/a) + C

Partial Fractions with Logs

∫ 1/(x²-a²) dx = (1/2a)ln|(x-a)/(x+a)| + C

Exponential Growth/Decay

∫ e^(kt) dt = e^(kt)/k + C

⚠️ Common Transcendental Mistakes

Chain Rule Errors

d/dx [e^(x²)] = 2x·e^(x²) NOT e^(x²)

Log Domain

ln(x) only for x>0. ln|x| for x≠0

Inverse Trig Range

sin⁻¹(x) gives values in [-π/2, π/2]

Exponential vs Power

d/dx [eˣ] = eˣ but d/dx [xⁿ] = n·xⁿ⁻¹

💪 Quick Practice Problems
1. d/dx [e^(3x)·ln(x)]

Answer: 3e^(3x)ln(x) + e^(3x)/x

2. ∫ sin(x)·e^(cos(x)) dx

Answer: -e^(cos(x)) + C

3. d/dx [tan⁻¹(2x)]

Answer: 2/(1+4x²)

4. ∫ x·e^(x²) dx

Answer: (1/2)e^(x²) + C