Teach 180: The Derivative of e^x (Day 104)

I had to have a substitute cover my last period class today and I never count on a sub to know math, especially Calculus.  Less damage control needs to be done the next day, if I go with that assumption.  Today's lesson was finding the derivative of f(x) = e^x and using the chain rule to find the derivative of g(x) = e^f(x) .  In years past, I had students use their graphing calculators to find the value of the derivative of f(x) = e^x  at various values of x.  Next, we would plot points such as (0, 1), (1, 2.718), (2, 7.4) and so on, and students would see that these points lined up on the function f(x) = e^x.

Since I wasn't going to be in class, I couldn't easily lead the students through this activity, and I had no expectation that my sub would be able to do this.  So, I created a short 8 minute video instead and posted it to my YouTube channel to show students why the derivative of f(x) = e^x is f(x) = e^x by using the limit definition of the derivative and a little help from desmos.  With desmos we could easily see the value of the following limit was 1.

To convince the students further, I used desmos to create a table of values for f(x) = e^x

and g(x) = f ' (x).  Behold, they have the exact same values!  Why?!?  Because they are the exact same function!

It's hard to believe that I started my YouTube channel back in November of 2008, almost 10 years ago.  Prior to Desmos, YouTube and Screencast-o-matic my sub plans would have consisted of students reading the book and working through examples.  Technology has certainly made my math lessons richer and has allowed for me to help students draw more connections among various representations.