Explore the shape of the gradient function for various exponential functions [math]y = a^x[/math].[br]The graph of [math]y = a^x[/math] is shown, a point P on the curve is shown with the tangent at that point. The gradient of the tangent is measured, and the value of the gradient is plotted at Q.

Choose a value for a.[br]Press the "trace Q" button.[br]Use the slider to adjust P.[br][br]1. What do you notice about the shape of the gradient function of [math]y = a^x[/math]?[br][br]Use the slider to set P to be 0.[br]Press "don't trace Q".[br]Use the slider to adjust a.[br][br]2. There is an important value of [math]a[/math] which leads to a special result. Write down the important value of [math]a[/math] as accurately as you can.[br][br]Keep a on this important value.[br]Press "Trace Q".[br]Use the slider to adjust P.[br][br]3. What do you notice about the gradient function of [math]y = a^x[/math] for this important value of [math]a[/math]?