When to use logarithmic differentiation:

1. y = [f(x)]^(g(x)) — variable base AND variable exponent
2. y = product of many functions (simplifies using log properties)
3. y = complicated quotient of products

Method for f(x)^g(x):

1. Take ln both sides: ln y = g(x) * ln(f(x))
2. Differentiate both sides: \frac{1}{y}$$\frac{dy}{dx} = g'(x)*ln(f(x)) + g(x)*f'$\frac{x}{f}$(x)
3. Multiply both sides by y: dy/dx = y * [g'(x)*ln(f(x)) + g(x)*f'$\frac{x}{f}$(x)]

Example: y = $x^x$ ln y = xln x \frac{1}{y}$$\frac{dy}{dx} = ln x + x$\frac{1}{x}$ = ln x + 1 dy/dx = $x^x$ * (1 + ln x)

Example: y = x^(sin x) ln y = sin x * ln x \frac{1}{y}$$\frac{dy}{dx} = cos x * ln x + sin x / x dy/dx = x^(sin x) * [cos x * ln x + sin x / x]

Common mistake: Trying to use the power rule d/dx($x^n$) = nx^(n-1) when the exponent is not a constant. The power rule works ONLY for constant exponents. For variable exponents, ALWAYS use logarithmic differentiation.