# 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 $\left|\underline{f}(b)-\underline{f}(a)\right|\le (b-a)\left|\underline{f'}(x)\right|$.
• msc/real_analysis_notes_by_syed_gul_shah/differentiation