====== 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’s 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 $\left|\underline{f}(b)-\underline{f}(a)\right|\le (b-a)\left|\underline{f'}(x)\right|$.

==== Download or View Online ====
<WRAP center round download 60%>
\\
**[[pdf>files/msc/real_analysis/dn.php?file=chap_04_real_analysis.pdf|Download PDF]]** (142KB) %%|%% **[[viewer>files/msc/real_analysis/chap_04_real_analysis|View Online]]**
</WRAP>