In today's class, I am confused about the following chain rule:
$$\frac{d}{dt}f(x(t),t) = f_x\frac{dx}{dt}+f_t$$
where $$f_t = \frac{\partial f}{\partial t}$$
What is the difference between $f_t$ and $\frac{d}{dt}f(x(t),t) $? It looks similar to me.
Can anyone come up with an concrete example to illustrate it?
Thanks so much
Consider the function $f(x,y)$ where $x=x(t),y=y(t).$ Then, it is
$$\dfrac{d}{dt}f(x(t),y(t))=f_x\dfrac{dx}{dt}+f_y\dfrac{dy}{dt}.$$
Now, assume $y(t)=t.$ Then $f_t$ means $f_y.$
Consider an example: $f(x,y)=x^2y, x(t)=t^2, y(t)=t.$ We have that
$$f_t=f_y=x^2=t^4$$ and
$$\dfrac{d}{dt}f(x(t),y(t))=2xy+x^2=2t^3+t^4.$$