Geometrically, you can view the derivative of a function at a point as the slope of the tangent line to the graph of the function at that point.[br][br]In the app below, move point [math]T[/math] along the graph of [math]f\left(x\right)=e^x[/math] and compare the value of the function at [math]T[/math], that is [math]e^{x_T}[/math], with the slope of the tangent line at [math]T[/math].[br]Do you notice anything interesting?
Considering the displayed values in the table of the app above, can you make a conjecture about what is the derivative of the function [math]f\left(x\right)=e^x[/math]?
Verify your conjecture by calculating the derivative of [math]f\left(x\right)=e^x[/math] using the limit of the difference quotient.[br][br][br][br]
The app below works as the previous one, but in this case we have the function [math]f\left(x\right)=e^{-x}[/math].[br][br]If you move point [math]T[/math] along the graph of this new function and compare the value of the function at [math]T[/math], that is [math]e^{-x_T}[/math], with the slope of the tangent line at [math]T[/math], do you notice anything interesting?
Considering the displayed values in the table of the app above, can you make a conjecture about what is the derivative of the function [math]f\left(x\right)=e^{-x}[/math]?
Verify your conjecture by calculating the derivative of [math]f\left(x\right)=e^{-x}[/math] using the limit of the difference quotient.