Chapter 04 - Differentiation

  • Derivative of a function
  • Theorem: Let f be defined on [a,b], if f is differentiable at a point $x\in [a,b]$, then f is continuous at x. (Differentiability implies continuity)
  • Theorem (derivative of sum, product and quotient of two functions)
  • Theorem (Chain Rule)
  • Examples
  • Local Maximum
  • Theorem: Let f be defined on [a,b], if f has a local maximum at a point $x\in [a,b]$ and if $f'(x)$ exist then $f'(x)=0$. (The analogous for local minimum is of course also true)
  • Generalized Mean Value Theorem
  • Geometric Interpretation of M.V.T.
  • Lagrange‚Äôs M.V.T.
  • Theorem (Intermediate Value Theorem or Darboux,s Theorem)
  • Related question
  • Riemann differentiation of vector valued function
  • Theorem: Let f be a continuous mapping of the interval [a,b] into a space $\mathbb{R}^k$ and $\underline{f}$ be differentiable in (a,b) then there exists $x\in (a,b)$ such that $\left|\underline{f}(b)-\underline{f}(a)\right|\le (b-a)\left|\underline{f'}(x)\right|$.

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