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Is the quotient rule applicable when dealing with differetiating functions of vectors?

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The expression to be differentiated has to make sense in the first place; e.g., it should not be a quotient of vectors. Assume this is the case. For instance, we are given a function

$h: \ {\mathbb R}^n\ \to\ {\mathbb R}, \quad h(x)\ := {f(x)\over g(x)}$

with real-valued $f$ and $g$, and we want to compute its gradient $\nabla h$. The $i$th component of this gradient is obtained by differentiating the quotient ${f\over g}$ with respect to the single real variable $x_i$, and for this operation the quotient rule is certainly valid:

$\bigl(\nabla h\bigr)_i(x)={\partial \over \partial x_i}\Bigl( {f(x)\over g(x)}\Bigr)= {f_{.i}(x)g(x)-f(x)g_{.i}(x) \over g^2(x)}\qquad(1\leq i\leq n)\ .$

In vectorial notation we therefore have

$\nabla h(x)={g(x)\ \nabla f(x)\ -\ f(x)\ \nabla g(x) \over g^2(x)}\ ,$

or

$\nabla{f\over g}={g\ \nabla f\ -\ f\ \nabla g\over g^2}\ ,$

which looks precisely like the familiar quotient rule.