Exponential & Logarithmic Derivatives

📅 Updated: November 23, 2024 ⏱️ 17 min read 📈 Complete Guide

Master exponential and logarithmic derivatives including e^x, a^x, ln(x), and log(x). Learn proofs, advanced techniques like logarithmic differentiation, and solve 25+ examples.

📑 Table of Contents

  1. Overview & Quick Reference
  2. Exponential Functions (e^x, a^x)
  3. Logarithmic Functions (ln, log)
  4. Why These Derivatives Work
  5. Solved Examples
  6. Logarithmic Differentiation
  7. Real-World Applications
  8. Tips & Common Mistakes
  9. Frequently Asked Questions

📊 Overview & Quick Reference

Exponential and logarithmic functions are fundamental in calculus, appearing everywhere from population growth to radioactive decay, compound interest to signal processing.

d/dx[e^x]
= e^x

The only function equal to its own derivative!

d/dx[a^x]
= a^x · ln(a)

For any constant base a > 0

d/dx[ln(x)]
= 1/x

Natural logarithm, x > 0

d/dx[log_a(x)]
= 1/(x·ln(a))

Logarithm with base a

Function Derivative Notes
e^x e^x Unique property: f = f'
e^(kx) k·e^(kx) Chain rule with constant k
a^x a^x·ln(a) General exponential base
ln(x) 1/x Domain: x > 0
ln|x| 1/x Works for x ≠ 0
log_a(x) 1/(x·ln(a)) Convert to ln: log_a(x) = ln(x)/ln(a)
⭐ Why e is Special

The number e ≈ 2.71828... is called Euler's number. It's the ONLY base where the exponential function equals its own derivative. This makes e the natural base for exponential growth and appears throughout mathematics, physics, and engineering.

📈 Exponential Functions

1. The Natural Exponential: e^x

🌟 d/dx[e^x] = e^x

The Most Important Derivative: e^x is its own derivative. This unique property makes it fundamental to calculus.

Basic Examples: d/dx[e^x] = e^x d/dx[3e^x] = 3e^x d/dx[e^x + x^2] = e^x + 2x With Chain Rule: d/dx[e^(2x)] = e^(2x) · 2 = 2e^(2x) d/dx[e^(x²)] = e^(x²) · 2x = 2x·e^(x²) d/dx[e^(-x)] = e^(-x) · (-1) = -e^(-x) d/dx[e^(3x+1)] = e^(3x+1) · 3 = 3e^(3x+1)

📝 Proof: Why d/dx[e^x] = e^x

Using the limit definition: d/dx[e^x] = lim[h→0] (e^(x+h) - e^x) / h = lim[h→0] (e^x · e^h - e^x) / h = lim[h→0] e^x(e^h - 1) / h = e^x · lim[h→0] (e^h - 1) / h Key fact: lim[h→0] (e^h - 1)/h = 1 = e^x · 1 = e^x ✓

2. General Exponential: a^x

📐 d/dx[a^x] = a^x · ln(a)

For any positive constant base a (where a ≠ 1), multiply the function by the natural log of the base.

Examples: d/dx[2^x] = 2^x · ln(2) ≈ 0.693 · 2^x d/dx[10^x] = 10^x · ln(10) ≈ 2.303 · 10^x d/dx[3^(2x)] = 3^(2x) · ln(3) · 2 = 2ln(3) · 3^(2x) d/dx[(1/2)^x] = (1/2)^x · ln(1/2) = -ln(2) · (1/2)^x Note: When a = e: d/dx[e^x] = e^x · ln(e) = e^x · 1 = e^x ✓
💡 Converting Bases

Any exponential can be written using e: a^x = e^(x·ln(a))

This is why the derivative includes ln(a)! Example: 2^x = e^(x·ln(2))

Properties of Exponential Derivatives

Key Properties:

  • e^x grows faster than any polynomial as x → ∞
  • e^(-x) decays to 0 as x → ∞ (exponential decay)
  • Chain rule always applies when exponent contains a function
  • Product/quotient rules apply when e^x is multiplied or divided

📉 Logarithmic Functions

1. Natural Logarithm: ln(x)

🌲 d/dx[ln(x)] = 1/x

Domain: x > 0 (ln is only defined for positive numbers)

Basic Examples: d/dx[ln(x)] = 1/x d/dx[5ln(x)] = 5/x d/dx[ln(x) + x] = 1/x + 1 With Chain Rule: d/dx[ln(2x)] = 1/(2x) · 2 = 1/x d/dx[ln(x²)] = 1/x² · 2x = 2/x d/dx[ln(3x+1)] = 1/(3x+1) · 3 = 3/(3x+1) d/dx[ln(sin(x))] = 1/sin(x) · cos(x) = cot(x)

📝 Proof: Why d/dx[ln(x)] = 1/x

Method 1: Using inverse functions Since ln and e are inverses: e^(ln(x)) = x Differentiate both sides: d/dx[e^(ln(x))] = d/dx[x] Using chain rule on left: e^(ln(x)) · d/dx[ln(x)] = 1 x · d/dx[ln(x)] = 1 d/dx[ln(x)] = 1/x ✓ Method 2: Limit definition d/dx[ln(x)] = lim[h→0] [ln(x+h) - ln(x)] / h = lim[h→0] ln[(x+h)/x] / h = lim[h→0] (1/h)ln(1 + h/x) = (1/x) lim[h→0] ln(1 + h/x)^(x/h) = (1/x) ln(e) = 1/x ✓

2. General Logarithm: log_a(x)

📐 d/dx[log_a(x)] = 1/(x·ln(a))

For logarithm with any base a, divide 1/x by ln(a).

Common Examples: d/dx[log₁₀(x)] = 1/(x·ln(10)) ≈ 0.434/x d/dx[log₂(x)] = 1/(x·ln(2)) ≈ 1.443/x d/dx[log₃(x)] = 1/(x·ln(3)) ≈ 0.910/x Conversion to natural log: log_a(x) = ln(x)/ln(a) Therefore: d/dx[log_a(x)] = d/dx[ln(x)/ln(a)] = (1/ln(a)) · d/dx[ln(x)] = (1/ln(a)) · (1/x) = 1/(x·ln(a)) ✓
✨ Special Case: ln|x|

d/dx[ln|x|] = 1/x for all x ≠ 0

The absolute value extends the domain to negative numbers! This is useful in integration and solving differential equations where x might be negative.

Logarithm Properties for Derivatives

Using Log Properties to Simplify:

Before differentiating, use log rules: ln(ab) = ln(a) + ln(b) ln(a/b) = ln(a) - ln(b) ln(a^n) = n·ln(a) Example: d/dx[ln(x³)] can be done two ways: Method 1 (chain rule): d/dx[ln(x³)] = (1/x³)·3x² = 3/x Method 2 (simplify first): ln(x³) = 3ln(x) d/dx[3ln(x)] = 3/x ✓ (easier!)

📝 Solved Examples

Example 1: Product with e^x

Find: d/dx[x²·e^x]

Using product rule: d/dx[uv] = u'v + uv' u = x², u' = 2x v = e^x, v' = e^x d/dx[x²·e^x] = 2x·e^x + x²·e^x = e^x(2x + x²) = x·e^x(2 + x) Answer: x·e^x(2 + x) or e^x·x(x + 2)
Example 2: Quotient with ln

Find: d/dx[ln(x)/x]

Using quotient rule: d/dx[u/v] = (u'v - uv')/v² u = ln(x), u' = 1/x v = x, v' = 1 d/dx[ln(x)/x] = [(1/x)·x - ln(x)·1] / x² = [1 - ln(x)] / x² Answer: (1 - ln(x))/x²
Example 3: Composite Functions

Find: d/dx[e^(sin(x))]

Using chain rule: Let u = sin(x), then du/dx = cos(x) d/dx[e^u] = e^u · du/dx d/dx[e^(sin(x))] = e^(sin(x)) · cos(x) Answer: cos(x)·e^(sin(x))
Example 4: ln of Composite

Find: d/dx[ln(x² + 1)]

Using chain rule: Let u = x² + 1, then du/dx = 2x d/dx[ln(u)] = (1/u) · du/dx d/dx[ln(x² + 1)] = 1/(x² + 1) · 2x = 2x/(x² + 1) Answer: 2x/(x² + 1)
Example 5: Mixed Functions

Find: d/dx[e^x·ln(x)]

Using product rule: u = e^x, u' = e^x v = ln(x), v' = 1/x d/dx[e^x·ln(x)] = e^x·ln(x) + e^x·(1/x) = e^x[ln(x) + 1/x] = e^x(ln(x) + 1/x) Answer: e^x(ln(x) + 1/x)

🎯 Logarithmic Differentiation

Logarithmic differentiation is a powerful technique for differentiating complicated functions, especially those with:

  • Variables in both base and exponent (like x^x)
  • Complicated products or quotients
  • Fractional or radical exponents
📋 Logarithmic Differentiation Steps
  1. Take ln of both sides

    If y = f(x), write ln(y) = ln(f(x))

  2. Simplify using log properties

    Use ln(ab) = ln(a) + ln(b), ln(a/b) = ln(a) - ln(b), ln(a^n) = n·ln(a)

  3. Differentiate both sides

    Use implicit differentiation: d/dx[ln(y)] = (1/y)·dy/dx

  4. Solve for dy/dx

    Multiply both sides by y and substitute original expression for y

Example 6: x^x (Variable Power)

Find: d/dx[x^x]

Step 1: Take ln of both sides y = x^x ln(y) = ln(x^x) = x·ln(x) Step 2: Differentiate both sides (1/y)·dy/dx = x·(1/x) + ln(x)·1 (1/y)·dy/dx = 1 + ln(x) Step 3: Solve for dy/dx dy/dx = y[1 + ln(x)] dy/dx = x^x[1 + ln(x)] Answer: x^x(1 + ln(x)) Verification at x = e: d/dx[x^x]|_{x=e} = e^e(1 + ln(e)) = e^e(1 + 1) = 2e^e
Example 7: Complicated Product

Find: d/dx[(x² + 1)³(x - 2)⁵]

Step 1: Take ln y = (x² + 1)³(x - 2)⁵ ln(y) = 3ln(x² + 1) + 5ln(x - 2) Step 2: Differentiate (1/y)·dy/dx = 3·(2x)/(x² + 1) + 5·1/(x - 2) (1/y)·dy/dx = 6x/(x² + 1) + 5/(x - 2) Step 3: Solve dy/dx = y[6x/(x² + 1) + 5/(x - 2)] dy/dx = (x² + 1)³(x - 2)⁵[6x/(x² + 1) + 5/(x - 2)] Can simplify to: dy/dx = (x² + 1)²(x - 2)⁴[6x(x - 2) + 5(x² + 1)]
Example 8: Variable Base and Exponent

Find: d/dx[(sin(x))^(cos(x))]

Step 1: Take ln y = (sin(x))^(cos(x)) ln(y) = cos(x)·ln(sin(x)) Step 2: Differentiate (product rule on right) (1/y)·dy/dx = -sin(x)·ln(sin(x)) + cos(x)·(cos(x)/sin(x)) (1/y)·dy/dx = -sin(x)·ln(sin(x)) + cos²(x)/sin(x) Step 3: Solve dy/dx = (sin(x))^(cos(x))[-sin(x)·ln(sin(x)) + cos²(x)/sin(x)] Answer: (sin(x))^(cos(x))[cos²(x)/sin(x) - sin(x)·ln(sin(x))]

🌍 Real-World Applications

📈 Exponential Growth & Decay

Population Growth: P(t) = P₀e^(rt)

Growth rate: dP/dt = rP₀e^(rt) = rP(t)

Radioactive Decay: N(t) = N₀e^(-λt)

Decay rate: dN/dt = -λN₀e^(-λt) = -λN(t)

💰 Compound Interest

Continuous Compounding: A(t) = Pe^(rt)

Where P = principal, r = rate, t = time

Rate of growth: dA/dt = rPe^(rt) = rA(t)

Example: Bacteria Growth

Bacteria population follows P(t) = 1000e^(0.3t) where t is hours.

Find growth rate at t = 5 hours: P(t) = 1000e^(0.3t) dP/dt = 1000·e^(0.3t)·0.3 = 300e^(0.3t) At t = 5: dP/dt = 300e^(1.5) ≈ 300(4.48) ≈ 1,344 bacteria/hour Interpretation: After 5 hours, bacteria are growing at 1,344 per hour

💡 Tips & Common Mistakes

❌ Mistake #1: Forgetting Chain Rule

Wrong: d/dx[e^(2x)] = e^(2x)

Right: d/dx[e^(2x)] = 2e^(2x)

Always multiply by the derivative of the exponent!

❌ Mistake #2: Confusing e^x and x^e

d/dx[e^x] = e^x (exponential function)

d/dx[x^e] = e·x^(e-1) (power function with constant exponent)

e^x has variable in exponent; x^e has variable in base!

❌ Mistake #3: Domain of ln(x)

Wrong: ln(-2) exists

Right: ln(x) only defined for x > 0

Use ln|x| if you need domain for x ≠ 0

❌ Mistake #4: Log Base Error

Wrong: d/dx[log₁₀(x)] = 1/x

Right: d/dx[log₁₀(x)] = 1/(x·ln(10))

Only ln(x) has derivative 1/x!

❓ Frequently Asked Questions

Q1: What is the derivative of e^x?

Answer: The derivative of e^x is e^x itself. This makes e^x unique - it's the only function that equals its own derivative. This property makes e fundamental in calculus and appears throughout mathematics, physics, and engineering. It's why e is called the "natural" base for exponentials.

Q2: Why is d/dx[ln(x)] = 1/x?

Answer: This comes from ln and e^x being inverse functions. Since they undo each other, their derivatives are related by the inverse function rule. Alternatively, it follows from the limit definition using properties of logarithms. The 1/x result means ln grows slowly - its rate of change decreases as x increases.

Q3: What is logarithmic differentiation and when should I use it?

Answer: Logarithmic differentiation means taking ln of both sides before differentiating. Use it for: (1) functions with variables in both base and exponent (like x^x), (2) complicated products or quotients, (3) when you want to avoid repeated product/quotient rules. It simplifies multiplication into addition and powers into multiplication.

Q4: How do I differentiate a^x for any base a?

Answer: d/dx[a^x] = a^x·ln(a). Multiply the function by the natural log of the base. For example: d/dx[2^x] = 2^x·ln(2). When a = e, ln(e) = 1, so d/dx[e^x] = e^x·1 = e^x. This formula works for any positive base a ≠ 1.

Q5: What's the difference between ln(x) and log(x)?

Answer: ln(x) is the natural logarithm (base e), while log(x) usually means base 10 in applied sciences (though mathematicians often use log for natural log). Their derivatives differ: d/dx[ln(x)] = 1/x, but d/dx[log₁₀(x)] = 1/(x·ln(10)) ≈ 0.434/x. Always check which base is meant!

Q6: Can I use logarithmic differentiation on any function?

Answer: Technically yes, but it's only helpful for certain types: products, quotients, powers with variables in exponents, or complicated fractional exponents. For simple functions like x² or sin(x), regular rules are faster. Logarithmic differentiation shines when multiplication/division is involved or when variables appear in exponents.

Q7: Why does e appear everywhere in calculus?

Answer: Because e is the unique number where e^x equals its own derivative. This makes exponential growth/decay with base e the "natural" rate of change. It appears in compound interest (continuous compounding), population growth, radioactive decay, normal distributions, Euler's formula (e^(iπ) + 1 = 0), and countless other applications. It's the base that makes calculus work smoothly!

Q8: How do I know when to use the chain rule with e^x or ln(x)?

Answer: Use chain rule whenever there's anything other than just "x" inside: e^(2x), e^(x²), ln(3x), ln(sin(x)) all need it. If it's just e^x or ln(x), no chain rule needed. Rule of thumb: if you can't substitute directly to evaluate, you need chain rule to differentiate!

🚀 Practice Exponential & Log Derivatives!

Use our free calculators to verify your derivatives and see step-by-step solutions!

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📚 Related Learning Resources

Derivative Rules Cheat Sheet

Quick reference for all derivative formulas

Chain Rule Explained

Essential for composite exp/log functions

Implicit Differentiation

Related to logarithmic differentiation

Trigonometric Derivatives

Learn trig function derivatives

Product Rule Guide

For multiplying exp/log with other functions

What is a Derivative?

Understand the fundamental concept

📌 Key Takeaways

  • ✅ d/dx[e^x] = e^x - The only function equal to its own derivative
  • ✅ d/dx[a^x] = a^x·ln(a) - Multiply by ln of the base
  • ✅ d/dx[ln(x)] = 1/x - Natural log derivative, x > 0
  • ✅ d/dx[log_a(x)] = 1/(x·ln(a)) - General log with base a
  • ✅ Always use chain rule when exponent or argument contains a function
  • ✅ Logarithmic differentiation simplifies products, quotients, and variable exponents
  • ✅ e ≈ 2.71828 is special because its exponential equals its derivative
  • ✅ ln and e^x are inverses - their derivatives are related
  • ✅ Use log properties to simplify before differentiating
  • ✅ Applications everywhere: growth, decay, interest, probability

Mastery Checklist

✓ Can you do these without notes?

  • □ Differentiate e^x, e^(2x), e^(x²)
  • □ Differentiate 2^x, 10^x, a^x
  • □ Differentiate ln(x), ln(2x), ln(x²)
  • □ Apply product rule with e^x or ln(x)
  • □ Apply quotient rule with exp/log functions
  • □ Use logarithmic differentiation on x^x
  • □ Differentiate complicated products using logs
  • □ Convert between different log bases
  • □ Explain why e^x = its own derivative
  • □ Solve real-world growth/decay problems

Study Strategy

  1. Master the four basic formulas first - e^x, a^x, ln(x), log_a(x)
  2. Practice chain rule extensively - it appears in most problems
  3. Learn logarithmic differentiation - powerful technique for hard problems
  4. Use log properties to simplify before differentiating when possible
  5. Understand the proofs - knowing WHY helps you remember
  6. Do 15 problems daily - mix basic and challenging
  7. Use our calculators to check your work and learn from mistakes