1. Exponential and Logarithmic Derivatives

2. Use your calculators to find the tangent line slopes to two exponential functions at x=0
y=3x
y=2x
mtan 
mtan  1.099
The question that arises from this is what base number would give us a slope of exactly 1 at x=0?

3. e  2.718 y = (2.718)x has a slope of almost 1 at x = 0
We find that the base number would be approximately…
 2.718
y = (2.718)x has a slope of almost 1 at x = 0
2.718… should look familiar to you as e 
So by the derivative definition, we know that…
Remember this!

4. Factor out an ex and we get
And since we know that
Which proves our easiest derivative to remember:
We now have

5. But what about a general base derivative?
First, recall these two equations from pre-calc:
and
Remember the chain rule and that ln a is a constant
And since we can substitute ax back in for this…

6. Use implicit differentiation to find

7. And so finally we have these four derivatives to remember