Question

\[\text{Let w(t) be a composite function: }\]\[\text{w(t)=f(X(t),Y(t))}\]\[\text{Derive the formula for the chain rule of w(t).}\]\[\text{You may use that near } x=x_{0}, y=y_{0} \text{,} \]\[f(x,y) \approx f(x _{0},y _{0}) + \frac{\delta f}{\delta x}(x_{0},y_{0})(x-x_{0}) + \frac{\delta f}{\delta y}(x_{0},y_{0})(y-y_{0})\]\[ \text{and linear Taylor's expansion of X(t), Y(t) near }t=t_{0} \]

Answer 1

\[ dw = \frac{\delta w}{\delta x} dx + \frac{\delta w}{\delta y} dy\]

Answer 2

\[ \frac{dw}{dt} = \frac{\delta w}{\delta x} \frac{dx}{dt} + \frac{\delta w}{\delta y} \frac{dy}{dt}\]

Answer 3

i think its asking you to drive the function, like f(X(t),Y(t)) = somthing

Answer 4

well ... the chain rule is above ... do you want to derive this relation??

Answer 5

\[ f(x(t), y(t))\approx f(x_0,y_0) + f'(x_0,y_0)x'(t_0)\Delta t + f'(x_0,y_0)y'(t_0)\Delta t \]

Answer 6

this might be silly but whats the \[\Delta t \] for?

Answer 7

\[ \Delta t = t - t_0\]

Answer 8

ahhh ok makes sense