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

Problem

Katherine wants to find the derivative of h, left parenthesis, x, right parenthesis, equals, minus, 18, natural log, left parenthesis, x, right parenthesis at the point x, equals, 3.
Her table below shows the slopes of the secant lines to the graph of h between the points left parenthesis, 3, comma, h, left parenthesis, 3, right parenthesis, right parenthesis and left parenthesis, x, comma, h, left parenthesis, x, right parenthesis, right parenthesis for x-values that get increasingly closer to 3:
xSlope of secant line, start fraction, h, left parenthesis, x, right parenthesis, minus, h, left parenthesis, 3, right parenthesis, divided by, x, minus, 3, end fraction
2, point, 9minus, 6, point, 1023
2, point, 99minus, 6, point, 0100
2, point, 999minus, 6, point, 0010
3, point, 001minus, 5, point, 9990
3, point, 01minus, 5, point, 9900
3, point, 1minus, 5, point, 9022
From the table, what does the derivative of h, left parenthesis, x, right parenthesis, equals, minus, 18, natural log, left parenthesis, x, right parenthesis at x, equals, 3 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