Question

If $u,v$ and $w$ are functions of $x$, then show that
$\frac{d}{dx}(u,v,w)=\frac{du}{dx}v.w+u.\frac{dv}{dx}.w+u.v\frac{dw}{dx}$
in two ways- first by repeated application of product rule, second by logarithmic differentiation.

Answer 1

Using Product rule, in which u and v are taken as one
$\frac{d\left(uvw\right)}{dx}$$=\frac{d\left(uv\right)}{dx}w+\frac{d\left(w\right)}{dx}uvw.v.\frac{d\left(u\right)}{dx}+w.u.\frac{d\left(v\right)}{dx}+u.v.\frac{d\left(w\right)}{dx}$
Now using logarithmic,
$y=uvw$
taking log on both sides, we have
$\log y=\log u+\log v+\log w\frac{1}{y}\frac{dy}{dx}=\frac{1}{u}\frac{du}{dx}+\frac{1}{v}\frac{dv}{dx}+\frac{1}{w}\frac{dw}{dx}\therefore \frac{dy}{dx}=y\times (\frac{1}{u}\frac{du}{dx}+\frac{1}{v}\frac{dv}{dx}+\frac{1}{w}\frac{dw}{dx})\frac{dy}{dx}=w.v.\frac{d\left(u\right)}{dx}+w.u.\frac{d\left(v\right)}{dx}+u.v.\frac{d\left(w\right)}{dx}$
since $y=uvw$