Section 3.13

Logarithmic Differentiation

When variables appear in both the base and the exponent, normal rules fail. Logarithms turn difficult products and powers into simple sums and products.

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Why Log Diff?

Power vs. Exponential

  • Power Rule: Base is variable, Exp is constant ().
  • Exponential Rule: Base is constant, Exp is variable ().
  • What about ? Neither rule works!

The Strategy

  1. Take of both sides.
  2. Use log properties to simplify.
  3. Differentiate strictly (implicit).
  4. Solve for .
2

Worked Example

Differentiate .

This is the classic case. The base is variable, and the exponent is variable.

Visualizing x^x

Defined for x > 0. It starts at 1, dips slightly, then explodes.

Original Function f(x)
Tangent Line
Derivative Function f'(x)
Step 1: Take Log
Step 2: Differentiate

Use Product Rule on the right:

Step 3: Solve for y'
3

Level Up Examples

Example A: Trig Exponents

Differentiate .

1. Log both sides:
2. Differentiate (Product Rule):
3. Multiply by y:

Example B: Algebraic Simplification

Differentiate using logs.

1. Expand using Log Properties:

Logs turn products to sums, quotients to differences, powers to coefficients.

2. Differentiate (Much easier!):
3. Multiply by y:
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Practice Quiz

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