Explore the algebraic form of the gradient function of exponential functions [math]y = a^x[/math] and meet THE exponential function, [math]e^x[/math].

Adjust the value of a using the slider.[br][br]1. What is the general form of the derivative of [math]y = a^x[/math]?[br]2. What do you notice about the function/graph [math]y = a^x[/math] and its derivative/gradient for the special value of [math]a[/math] found in Task 2?[br][br]The special value is called "[math]e[/math]".[br][br]3. Use your calculator to find the value of [math]e[/math] to 3 decimal places.[br][br][math]a^x[/math] is an exponential function. [math]e^x[/math] is THE exponential function.[br][br]Type "f(x)=e^x" into the input box, selecting "e" from the drop-down menu to the right of the box.[br][br]4. Write down the derivative of [math]e^x[/math].[br]5. Hence write down the integral of [math]e^x[/math].