Day 6 in MAT229

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Today: Derivatives of logs.
- I've updated your assignments and schedule for the week.
- A reminder that our first exam is coming up: not this weekend, but the next weekend. You'll have several days over which you can take it, but it will be by lockdown browser (and probably a couple of IMath problems).
  It's going to be a recap of some problems from Calc I, with a new collection of functions (these exponentials, and inverse functions).
- So the big picture is that we're looking into the properties of these log functions, and focusing on the natural log, $ln(x)$, because it corresponds to the natural exponential $e^x$.
  The natural exponential $e^x$ is its own derivative -- fantastic!
  The natural log $ln(x)$ has its own amazing property: \[ \frac{d}{dx}ln(x) = \frac{1}{x} = x^{-1} \]
  So we can now integrate that one power, that caused trouble when we tried to use the power rule: \[ \int x^{-1} {dx} = ln(|x|)+C \] Note the $|x|$:
  
  Then $ln(|x|)$ becomes an even function; its derivative is $\frac{1}{x}$, an odd function.
  Did you know about that symmetric relationship between a function and its derivative?
  What's true about the derivative of an odd function?
- Materials:
  - Log Derivatives (.nb)
  - Log Derivatives (.pdf)
  - Long yacks through the notes using Mathematica:
    - Part A
    - Part B

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