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Logarithmic & Exponential Differentiation Guide

Master the art of differentiating exponential and logarithmic functions with this comprehensive guide. Learn formulas, techniques, and practice problems.

📋 Table of Contents
📈
1. The Natural Exponential Function eˣ
The only function that equals its own derivative
⚡ Fundamental Property
d/dx [eˣ] = eˣ

Why is this special?

The function eˣ is the unique solution to the differential equation f'(x) = f(x) with f(0) = 1. This makes it incredibly important in mathematics, physics, and engineering.

🔗 Chain Rule Extension
d/dx [eᵘ] = eᵘ · u'

Where u = u(x)

This is the most common form you'll encounter. When differentiating e raised to any function, multiply by the derivative of that function.

📝 Worked Examples
1 d/dx [e³ˣ] = e³ˣ · 3 = 3e³ˣ
2 d/dx [e^(x²)] = e^(x²) · 2x = 2xe^(x²)
3 d/dx [e^(sin x)] = e^(sin x) · cos x
🔢
2. General Exponential Functions aˣ
Derivatives for any positive constant base a
📊 General Formula
d/dx [aˣ] = aˣ · ln(a)

Derivation: Using logarithmic differentiation:

1. Let y = aˣ

2. ln(y) = x·ln(a)

3. Differentiate: y'/y = ln(a)

4. Solve: y' = y·ln(a) = aˣ·ln(a)

🔗 With Chain Rule
d/dx [aᵘ] = aᵘ · ln(a) · u'

Special Cases:

• When a = e: ln(e) = 1, so formula reduces to eᵘ·u'

• When a = 10: ln(10) ≈ 2.3026

• When a = 2: ln(2) ≈ 0.6931

📝 Worked Examples
1 d/dx [2ˣ] = 2ˣ · ln(2)
2 d/dx [10^(x²)] = 10^(x²) · ln(10) · 2x
3 d/dx [πˣ] = πˣ · ln(π)
📐
3. Natural Logarithm ln(x)
The inverse function of eˣ
📖 Basic Derivative
d/dx [ln(x)] = 1/x

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

Derivation using inverse function theorem:

If y = ln(x), then x = eʸ

Differentiate: dx/dy = eʸ = x

Therefore: dy/dx = 1/x

🔗 With Chain Rule
d/dx [ln(u)] = u'/u

Where u = u(x) > 0

This is the most useful form. The derivative of ln(u) is simply the derivative of u divided by u.

Memory Aid: "Derivative over the function"

📝 Worked Examples
1 d/dx [ln(x³)] = (3x²)/(x³) = 3/x
2 d/dx [ln(sin x)] = cos x / sin x = cot x
3 d/dx [ln(√x)] = (1/(2√x))/(√x) = 1/(2x)
📊
4. General Logarithmic Functions
Logarithms with any base a > 0, a ≠ 1
🔢 Change of Base Formula
logₐ(x) = ln(x)/ln(a)

Using this formula:

We can convert any logarithm to natural logarithms, making differentiation easier.

This works because ln(a) is just a constant.

📐 Derivative Formula
d/dx [logₐ(x)] = 1/(x·ln(a))

Derivation:

Using change of base: logₐ(x) = ln(x)/ln(a)

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

Special Case: When a = e, ln(e) = 1, so we get 1/x

📝 Worked Examples
1 d/dx [log₁₀(x)] = 1/(x·ln(10))
2 d/dx [log₂(x³)] = (3x²)/(x³·ln(2)) = 3/(x·ln(2))
3 d/dx [log₅(sin x)] = cos x/(sin x·ln(5)) = cot x/ln(5)
5. Logarithmic Differentiation Technique

A powerful method for differentiating complex functions, especially those involving products, quotients, and powers.

1

Take Natural Log of Both Sides

Start with y = f(x). Take ln of both sides: ln(y) = ln(f(x))

Example: For y = xˣ, we get ln(y) = ln(xˣ) = x·ln(x)

2

Use Logarithm Properties to Simplify

Apply: ln(ab) = ln(a) + ln(b), ln(a/b) = ln(a) - ln(b), ln(aᵇ) = b·ln(a)

Example: For y = (x²+1)·eˣ/(x-1), we get: ln(y) = ln(x²+1) + x - ln(x-1)

3

Differentiate Implicitly

Differentiate both sides with respect to x. Remember: d/dx[ln(y)] = y'/y

Example: For ln(y) = x·ln(x), we get: y'/y = ln(x) + 1

4

Solve for y'

Multiply both sides by y: y' = y · (derivative of right side)

Example: y' = xˣ · (ln(x) + 1)

5

Substitute Original y if Needed

Replace y with the original expression if it simplifies the result

Example: y' = xˣ(ln x + 1) is already in simplest form

🎯 When to Use Logarithmic Differentiation
Products & Quotients

y = (x²+1)(x³-2)/√x

Functions to a Function

y = xˣ, y = (sin x)^x

Complex Combinations

y = √(x·eˣ/(x+1))

6. Practice Problems