What is $\frac{\partial(\frac{d(x(t))}{dt})}{(\partial x(t))}$?

Question

What is $$\frac{\partial(\frac{d(x(t))}{dt})}{(\partial x(t))}$$ ?

I came across this while trying to find the second derivative of $$u(t)=f(X(t),Y(t))$$.

Edit: When we are taking the second derivative, we have $$\frac{\partial}{\partial x} (\frac{\partial f}{\partial x} x' + \frac{\partial f}{\partial y} y') x' + \frac{\partial }{\partial y}(\frac{\partial f}{\partial x} x' + \frac{\partial}{\partial y} y') y'$$

but, for example, in the first part when we are to take the derivative of $\frac{\partial f}{\partial x} x'$ respect to x, we should get $\frac{\partial^2 f}{\partial x^2} x' + \frac{\partial f}{\partial x} \frac{\partial x'}{\partial x}$ but I don't know what $\frac{\partial x'}{\partial x}$ is .

Answer 1

We have

$df(a,b)=\frac{\partial f}{\partial x}(a,b)dx+\frac{\partial f}{\partial y}(a,b)dy$

hence

$$u'(t)=\frac{df(X(t),Y(t))}{dt}$$

$=f'_x(X(t),Y(t))X'(t)+f'_y(X(t),Y(t))Y'(t)$

and

$u''(t)=X'( f''_{xx}X'+f''_{xy}Y' )$

$+X''f'_x$

$+Y'(f''_{yx}X'+f''_{yy}Y')$

$+Y''f'_y$