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