Explore the form of the gradient function of exponential functions . The graph of 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.
Press the "trace Q" button.
Use the slider to adjust P.
1. What do you notice about the shape of the gradient function of ?
Use the slider to set P to be 0.
Press "don't trace Q".
Use the slider to adjust a.
2. There is an important value of which leads to a special result. Write down the important value of as accurately as you can.
Keep a on this important value.
Press "Trace Q".
Use the slider to adjust P.
3. What do you notice about the gradient function of for this important value of ?