The generalization of the derivative to functions of several variables is the gradient.Where the function $f$ depends on one or more variables $x \in R^n$, e.g.,$f(x) = f(x1,x2)$. We find the gradient of the function $f$ with respect to $x$ by varying one variable at a time and keeping the others constant. The gradient is then the collection of these partial derivatives. Definition Partial Derivative For a function $f: R^n \to R$, $x \to f(x), x \in R^n$ of $n$ variables $x_1,x_2,\ldots,x_n$, we define partial derivatives as $\frac{\partial f}{\partial x_1}=\lim_{h \to 0}\frac{f(x_1+h,x_2,\ldots,x_n)-f(x))}{h}$ $\vdots$ $\frac{\partial f}{\partial x_n}=\lim_{h \to 0}\frac{f(x_1,x_2,\ldots,x_n+h)-f(x))}{h}$ and collect them in a row vector.The row vector is called the gradient of $f$ or the Jacobian and is the generalization of the derivative form. Example: if $f(x1,x2)=x_1^2x_2+x_1x_2^3 \in R$, then the derivative of $f$ with respect to $x_1$ and $x_2$ are. $\frac{\parti...