Gradients on y = ax: the factor ln a comes out of the curve

Year 12 Mathematics Advanced · the gradient is measured from the curve, not looked up · the readout to watch is m ÷ y · axes x from −3 to 3, y from −2 to 10
Base a

Graph

y = ax tangent marks (x, m)
Drag the point along the curve, or use the x slider. Watch m ÷ y: it does not move when x moves.
1
2.00

a = 1 is a flat line, so the slider runs from 0.5 to 4 through it.

point (1, 2.00)
gradient m (measured) 1.39
m ÷ y 0.69
ln a hidden

m ÷ y against the base a

Change a, record the ratio, repeat. The points trace a curve you know.

Which candidate do the marks land on?

Place marks at a = 2, then switch all three on. Two are the lesson's traps.

Find the special base

Move a until m ÷ y reads exactly 1.00, so the curve is its own derivative.