simple differentation - help

Question

how do you differentiate the expression vt with respect to x i.e. (d/dx)(vt) ?

I know this is pretty simple but my mind has gone completely blank

any help would be greatly appreciated! thank you :)

Answer 1

Remember the product rule from calculus: If $f(x)$ and $g(x)$ are two differentiable functions of $x$, then $h(x) = f(x)g(x)$ is differentiable and its derivative is given by \begin{align} \frac{d}{dx}h(x) = \frac{d}{dx} (f(x)g(x)) = f(x)\frac{dg(x)}{dx}+\frac{df(x)}{dx}g(x). \end{align} I do not know how your problem is formulated, e.g. does $v$ and $t$ depend on $x$? Regardless, we have \begin{align} \frac{d}{dx}(vt) = t\frac{dv}{dx}+\frac{dt}{dx}v. \end{align} Then, for example if $t$ does not depend on $x$, then the first term will be zero, or if $v$ does not depend on $x$, then the second term will be zero.