Question

derivative of vector eith respect to scaler quantity is? sacler, vector or matrix?​

Answer 1

Answer:

1

Partial derivatives are directional (Gâteaux) derivatives, in other words, since w∈R

,

∂fw∂w=limh∈R,h→0fw+h−fwh

Therefore

∂fw∂w=limh→01h(w+h−w)⎛⎝⎜abc⎞⎠⎟=⎛⎝⎜abc⎞⎠⎟

Your f

is just a linear map! Its derivative is the constant function

f′:R→R3,x↦⎛⎝⎜abc⎞⎠⎟.

More generally if you have f

given as a function f=⎛⎝⎜f1f2f3⎞⎠⎟ where f1,f2,f3:R→R are differentiable, then the derivative of f will be ⎛⎝⎜f′1f′2f′3⎞⎠⎟.

Note that f⃗ =⎡⎣⎢awbwcw⎤⎦⎥

.

Now, differentiate each component of the vector with respect to w

. That is:

∂f⃗ ∂w=⎡⎣⎢⎢⎢∂∂waw∂∂wbw∂∂wcw⎤⎦⎥⎥⎥=⎡⎣⎢abc⎤⎦⎥

This assumes a,b,c

could be constants or functions (but not of w).

Explanation: