# A simple calculus question

## Main Question or Discussion Point

i just want simple explanation of this.
velocity v is a function of r and t.
then my professor, in order to derive relationship of something, he wrote:
$$\frac{dv}{dt}=\frac{\partial v}{\partial r}+\frac{\partial v}{\partial t}$$
then assuming small spatial variation,
$$\frac{dv}{dt} \approx \frac{\partial v}{\partial t}$$
my question: is the first equation above OK?

thanks.

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look up the chain rule; when v = v(r(t),t),

$$\frac{dv}{dt}=\frac{\partial v}{\partial r}\frac{dr}{dt}+\frac{\partial v}{\partial t}$$

then you can see the following argument makes more sense?

thanks. i just want to clarify again, does it mean dr/dt is small, i.e. small velocity?