Chapter 12.7, Problem 65E

Logarithmic differentials Let f be a differentiable function of one or more variables that is positive on its domain.

a.    Show that $d\left(\ln f\right)=\frac{df}{f}$.

b.    Use part (a) to explain the statement that the absolute change in ln f is approximately equal to the relative change in f.

c.    Let f(x, y) = xy, note that ln f = ln x + ln y, and show that relative changes add: that is, df/f = dx/x + dy/y.

d.    Let f(x, y) = x/y, note that ln f = ln x = ln y, and show that relative changes subtract; that is df /f = dx/x – dy/y.

e.    Show that in a product of n numbers, f = x₁x₂…x_n, the relative change in f is approximately equal to the sum of the relative changes in the variables.