$\begin{align}  & y=uvw \\ & y+\delta y=(u+\delta u)(v+\delta v)(w+\delta w) \\ & \delta y=(u+\delta u)(v+\delta v)(w+\delta w)-uvw \\ & \delta y=(uv+u\delta v+v\delta u+\delta u\delta v)(w+\delta w)-uvw \\ & \delta y=uvw+uw\delta v+vw\delta u+w\delta u\delta v+uv\delta w+u\delta u\delta w+v\delta u\delta w+\delta u\delta v\delta w-uvw \\ & \delta y=uw\delta v+vw\delta u+w\delta u\delta v+uv\delta w+u\delta u\delta w+v\delta u\delta w+\delta u\delta v\delta w \\ & \text{Dividing both sides by }\delta x \\ & \frac{\delta y}{\delta x}=uw\frac{\delta v}{\delta x}+vw\frac{\delta u}{\delta x}+w\frac{\delta u\delta v}{\delta x}+uv\frac{\delta w}{\delta x}+u\frac{\delta u\delta w}{\delta x}+v\frac{\delta u\delta w}{\delta x}+\frac{\delta u\delta v\delta w}{\delta x} \\ & \text{Taking limits as }\delta x\to 0\text{ and hence }\delta y,\delta u,\delta v,\delta w\to 0 \\ & \frac{dy}{dx}=\underset{\delta x\to 0}{\mathop{\lim }}\,\left( uw\frac{dv}{dx}+vw\frac{du}{dx}+w(0)\frac{dv}{dx}+uv\frac{dw}{dx}+u(0)\frac{dw}{dx}+v(0)\frac{dw}{dx}+(0)(0)\frac{dw}{dx} \right) \\ & \frac{dy}{dx}=uw\frac{dv}{dx}+vw\frac{du}{dx}+uv\frac{dw}{dx} \\\end{align}$