ddxln u= u u (u>0) The logarithm and natural exponential undo each other; a small relative input change gives a logarithmic change, giving the reciprocal derivative. Composition multiplies by the inner rate. Method Require the logarithm argument to be positive. Differentiate the argument, divide by it, simplify, and preserve that domain. Example y=ln(9-x2), dydx= -2x9-x2, -3<x<3 Exam check The derivative denominator being nonzero is insufficient: the original logarithm must also be defined.

\(\dfrac{\mathrm{d}}{\mathrm{d}x}\ln u=\dfrac{u^{\prime}}{u}\quad(u>0)\)

The logarithm and natural exponential undo each other; a small relative input change gives a logarithmic change, giving the reciprocal derivative. Composition multiplies by the inner rate.

Require the logarithm argument to be positive. Differentiate the argument, divide by it, simplify, and preserve that domain.

\(y=\ln(9-x^2),\quad\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{-2x}{9-x^2},\quad-3<x<3\)

The derivative denominator being nonzero is insufficient: the original logarithm must also be defined.

Natural logarithm derivatives

Natural logarithm derivatives