Day 18, Mat128: Calculus A

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Announcements
- Your quizzes are returned.
- Your 2.2 homework is due today.
We continue with Section 2.3: Differentiation formulas
- You remember how ugly the algebra was to use the definition to find the derivative of that radical function from last time? That was nasty, wasn't it? Wouldn't it be nice if we could avoid all of that? Well we can, using some rules that are derived from the limit definition of the derivative.
- So now we'll use the definition of the derivative to prove several of the formulas in this section. We've already proven the first two rules (constant functions $f(x)=c$ and the linear function $f(x)=x$).
  - Rules that help us avoid having to use the definition each time (with Proofs):
    - The constant multiple rule (makes sense)
      
      $(cf(x))'=cf'(x)$
    - The sum rule
      works as we'd hope: "The derivative of a sum is the sum of the derivatives."
      
      $(f(x) + g(x))'=f'(x)+g'(x)$
    - The power rule (via dominoes -- i.e., mathematical induction -- or via the binomial theorem).
      The binomial theorem is another algebra problem for folks, and provides another way to pull a rabbit out of my ... hat!
      
      $(x^n)'=n x^{n-1}$
    - NOTE: at this point we have all the rules necessary to differentiate all polynomials, without needing to resort to the definition!
      The derivative of the monomial $x^n$ is $nx^{n-1}$, and
      The derivative of the monomial $c x^n$ is $nc x^{n-1}$ (by constant multiple).
      A polynomial is just a sum of these. So we apply the sum rule, and the power rule, and the constant multiple rule to the flight of the eraser, to get
      $s'(t)=(at^2+bt+c)'=2at+b$
      and
      $s''(t)=(2at+b)'=2a$
  - Other rules:
    - The product rule (doesn't work the way we'd hope):
      
      $(f(x) \cdot g(x))'=f'(x)g(x)+f(x)g'(x)$
    - The quotient rule:
      
      low d high minus high d low,
      over the denominator squared they go.
      $\left(\frac{f(x)}{g(x)}\right)'=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}$
- Examples:
  - #13, p. 136
  - #22, p. 136 (expand first)
  - #48, p. 137
  - #58, p. 137

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