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

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