Partial derivative of product of vector transpose and vector.

Question

While watching "CS 224D Lecture 2" at 41:51 the professor writes on the board:

$$\frac{\partial }{\partial v_c}u^T_ov_c=u_o$$

I initially thought the result was going to be :

$$\frac{\partial }{\partial v_c}u^T_ov_c=u^T_o$$

Which seems to be the expected result as you are treating $u^T_o$ as a constant.

I searched for other instances of this formula and at "Derivatives with respect to vectors" Example 3, the author confirms:

$$\frac{\partial }{\partial x}u^Tx=u^T$$

While not reliable, Wolfram Alpha also gives the above formula.

Question: Why does the Transpose "disappear" in the first formula?

Thanks in advance.

Answer 1

This is a relaxation of notation that is causing your confusion. $$\frac{\partial f}{\partial x}$$ when $x$ is a vector means the vector with the $i$th component $$\frac{\partial f}{\partial x_i}$$ Also, remember $a^Tb$ is shorthand for $$\sum_i a_i b_i$$ so that $$\frac{\partial}{\partial x}a^Tx = a$$