Logarithmic Differentiation

Use when differentiating complex products, quotients, or variable bases with variable exponents.

Method:

1. Take ln of both sides: ln y = ln[f(x)]

2. Use log rules to simplify

3. Differentiate both sides (implicit differentiation on left)

4. Solve for dy/dx

Essential for: y = xˣ or y = (sin x)^(cos x)

y = xˣ → ln y = x ln x → (1/y)(dy/dx) = ln x + 1

dy/dx = xˣ(ln x + 1)

Also useful for simplifying:

y = (x²(x+1)³)/(x-2)⁴

ln y = 2ln x + 3ln(x+1) - 4ln(x-2)

Much easier to differentiate this sum than the original product/quotient.

Reference:

Wikipedia: Log Differentiation

https://en.wikipedia.org/wiki/Logarithmic_differentiation