Logarithmic differentials Let f be a differentiable function of one or more variables that is positive on its domain.
a. Show that
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…xn, the relative change in f is approximately equal to the sum of the relative changes in the variables.
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