Logarithmic differentiation

For y = [u(x)]^(v(x)) or a long product and quotient, take log first: log y turns powers into products and products into sums. Then differentiate implicitly and multiply by y.

It needs f(x) > 0 for log to be defined. The power rule nxⁿ⁻¹ is only for a constant exponent; xˣ needs logs.

Worked example: y = aˣ, a > 0. log y = x log a, so y′/y = log a and d/dx(aˣ) = aˣ log a.

Worked example: y = x^(sin x), x > 0. log y = sin x log x, so y′/y = cos x log x + (sin x)/x, and y′ = x^(sin x)[cos x log x + (sin x)/x], which is x^(sin x − 1) sin x + x^(sin x) cos x log x.

Worked example: y = √[(x − 3)(x² + 4)/(3x² + 4x + 5)]. Taking logs splits it into ½[log(x − 3) + log(x² + 4) − log(3x² + 4x + 5)], and each piece is differentiated separately.

Sums of such powers: for yˣ + xʸ + xˣ = aᵇ, differentiate each term separately by logs; the right side is a constant with derivative 0.

Worked example: xˣ. log y = x log x gives y′/y = log x + 1, so d/dx(xˣ) = xˣ(1 + log x).