Find derivatives of functions by considering them as the limits of secant line slopes or average rate of change.

Problem

Jude wants to find the derivative of h, left parenthesis, x, right parenthesis, equals, 12, natural log, left parenthesis, x, right parenthesis at the point x, equals, 4.
His table below shows the average rate of change of h over the intervals open bracket, x, comma, 4, close bracket or open bracket, 4, comma, x, close bracket for x-values that get increasingly closer to 4:
xIntervalAverage rate of change, start fraction, h, left parenthesis, x, right parenthesis, minus, h, left parenthesis, 4, right parenthesis, divided by, x, minus, 4, end fraction
3, point, 9open bracket, 3, point, 9, comma, 4, close bracket3, point, 0381
3, point, 99open bracket, 3, point, 99, comma, 4, close bracket3, point, 0038
3, point, 999open bracket, 3, point, 999, comma, 4, close bracket3, point, 0004
4, point, 001open bracket, 4, comma, 4, point, 001, close bracket2, point, 9996
4, point, 01open bracket, 4, comma, 4, point, 01, close bracket2, point, 9963
4, point, 1open bracket, 4, comma, 4, point, 1, close bracket2, point, 9631
From the table, what does the derivative of h, left parenthesis, x, right parenthesis, equals, 12, natural log, left parenthesis, x, right parenthesis at x, equals, 4 appear to be?
  • Your answer should be
  • an integer, like 6
  • an exact decimal, like 0, point, 75
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4