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Section 5.3 Limits and differentiation

Introduction goes here.
  • Limit definition of a derivative:
    \begin{align*} f'(a) \amp = \limit{x\to a}\frac{f(x)-f(a)}{x-a}\\ f'(x) \amp = \limit{h\to 0}\frac{f(x+h)-f(x)}{h} \end{align*}
  • Work some examples of limits to show they still work out to the known derivatives
  • Prove Product Rule and Chain Rule
  • \(0/0\) as an indeterminate form
  • Ways in which a function can fail to be differentiable, using \(|x|\) as the example:
    \begin{equation*} \dv{x}|x|=\frac{|x|}{x} \end{equation*}
  • Role of differentials as changes along the tangent line