Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am trying to understand the basic rules of vector differentiation.

In a non-scalar expression where x is a column vector, is it valid to differentiate with respect to the kth element of x ?

For example, what would be the result of the following?

$ \partial/\partial x_k (x^T) $

$ \partial/\partial x_k (x) $

share|cite|improve this question
up vote 1 down vote accepted

Indeed, if you consider a vector $x$ to be a function of its entries. Then we have $$\frac{\partial}{\partial x_k}x = \begin{pmatrix} 0\\ \vdots\\ 1\\ 0\\ \vdots\\ 0\end{pmatrix} \text{ and }\frac{\partial}{\partial x_k}x^T = \begin{pmatrix} 0, \ldots, 1, 0, \ldots,0\end{pmatrix}$$ where the $1$ is in the $k^{th}$ spot. More generally, there is a notion of derivative for functions between any two Banach spaces (which includes $\mathbb R^n$ for all $n$), under which the derivative of a function $f$ at a point $a$ is the unique linear function $A$ such that $$\lim\limits_{x\to a} \frac{\|f(x)-f(a)-A(x-a)\|}{\|x-a\|}=0$$ and this agrees with the notion of a derivative from Calculus I because when $f$ is a function from $\mathbb R$ to $\mathbb R$ we get a linear function $cx$, and the constant $c$ is precisely the derivative (in the Calc I sense) of $f$ at $a$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.