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