Parametric equations and derivatives

[Shaybay92]

Just a quick question... if we have f(x,y,z) and x(t), y(t), z(t), without substituting in what x y and z are in f, how do we calculate df/dt?

 

[I like Serena]

Just a quick question... if we have f(x,y,z) and x(t), y(t), z(t), without substituting in what x y and z are in f, how do we calculate df/dt?

$$\frac {df} {dt} = \frac {\partial f} {\partial x} \frac {dx} {dt} + \frac {\partial f} {\partial y} \frac {dy} {dt} + \frac {\partial f} {\partial z} \frac {dz} {dt}$$

 

[Shaybay92]

Where did this come from? I can't see why we should add the contributions of each?

 

[I like Serena]

Where did this come from? I can't see why we should add the contributions of each?

This is the standard formula for multiple-variable derivatives, also called the "total derivative".

In words: if t would increase a little bit, x, y, and z each will change a little bit as well.
This will make f change a little bit as well. There will be a partial change caused by the change in x, and also a partial changed caused by the change in y.
All in all, all the partial changes need to be added.

This is the reason the "round d" is used in ∂f/∂x to signify it's about the partial change of f, due to a change in x. This needs to be multiplied by the change that x takes due to the change in t. Since the last is not a "partial" change, a "regular d" is used as in dx/dt.

 

[Shaybay92]

Thankyou!