Consider the applet above. The main function *f(x)* is in blue, the tangent line is in green, and the derivative function *f '(x)* is graphed in red.
- Identify the similarities between functions of form
*x*^{n}, such as appear in the applet XToN, and of form *a*^{x}, such as appear in the applet below. It might help to consider specific examples such as *x*^{3} & *3*^{x} or *x*^{(1/2)}, & *(1/2)*^{x}.
- Identify at least 3 differences between functions of form
*x*^{n} and of form *a*^{x}. Again, it might help to consider specific examples.
- Set
*f(x)=2*^{x} using the slider on the right side of the applet. Assume for the moment that the derivative of *f(x)=2*^{x} is *f '(x)=x 2*^{x-1}
`a.` Given the assumption, find the derivative of *f(x)=2*^{x} at *x=-1, 0,* and *1*.
`b.` Using the applet, estimate the actual derivative of *f(x)=2*^{x} at *x=-1, 0,* and *1*.
`c.` What do (**a** & **b**) tell us about our assumption that the derivative of *f(x)=2*^{x} is *f '(x)=x 2*^{x-1}? Why?
`d.` Since we cannot use the Power Rule to take the derivative of *f(x)=a*^{x}, how can we find the derivative?

- Set
*f(x)=2*^{x} using the slider on the right side of the applet. It appears that the derivative of *f(x)=2*^{x} is just *2*^{x} times some constant. Assuming this, estimate the constant. (Hint: your answers to **3b** may be helpful.)
- Set
*f(x)=3.7*^{x} using the slider on the right side of the applet. It appears that the derivative of *f(x)=3.7*^{x} is just *3.7*^{x} times some constant. Assuming this, estimate the constant.
- Is there any number
*a* so that the derivative of *f(x)=a*^{x} is just *a*^{x} (that is, the constant in a question similar to (**4** or **5**) is just 1)? If so, estimate *a*. If not, explain why not.