\part{Differentiation}
- differentiable function
- derivative
- product rule
- chain rule
- quotient rule
- maxima
- Rolle's theorem
- mean value theorem (MVT)
- generalized mean value theorem (GMVT)
- L"Hopital's rule
- inverse function theorem
- convex function
- concave funtion
- concave funtion
- global maximum
- global minimum
- local maximum
- local minimum 
- power series
- Taylor series
- Taylor's theorem
- analytic function
- C^k spaces
- smooth function
- continuously differentiable function
\end{definition}
- Hessian matrix
\begin{definition}
The \emph{Hessian matrix of $f$} is the matrix of second-order partial derivatives.
\[ \mathbf{J}_{f} \]
\[ (\mathbf{J}_{f})_i = \frac{\partial f}{\partial x_i} \]
\end{definition}
\begin{proposition}
Let $f$ have continuous second-order partial derivatives on $U$. Then $f$ is convex on $U$ iff its Hessian matrix $\mathbf{H}_{f}$ is positive semi-definite.
\end{proposition}
- first derivative test
- second derivative test
- Multidimentional Taylor series
- Multidimentional aylor's theorem
- Multidimensional second derivative test
