Past exam of the options I course of the University of Oxford / 2013 / F / 1 / a

The total derivative of $f$ at $x$ is a linear map $Df(x):{\mathbb{R}}^n\to \mathbb{R}$ satisfyingThe gradient is the vector of partial derivatives, $\nabla f(x)=({\partial }_{1}f(x),\dots ,{\partial }_{n}f(x))$. When $f$ is totally differentiable, these partial derivatives exist andThus the derivative is a linear functional represented by the gradient under the Euclidean inner product. Existence of partial derivatives alone does not imply total differentiability.

The denominator is positive on $B$ because $x_3>-1$. The quotient rule gives the gradientIn particular $g$ is continuously differentiable throughout $B$.