Answer
See explanations.
Work Step by Step
a. Step 1. Recall the Derivative Product Rule: $\frac{d}{dx}(uz)=u\frac{dz}{dx}+z\frac{du}{dx}$
Step 2. Let $z=vw$, we have $\frac{d}{dx}(z)=\frac{d}{dx}(vw)=v\frac{dw}{dx}+w\frac{dv}{dx}$
Step 3. Combine the above results: $\frac{d}{dx}(uz)=\frac{d}{dx}(uvw)=u(v\frac{dw}{dx}+w\frac{dv}{dx})+vw\frac{du}{dx}$
Step 4. Rewrite the above results: $\frac{d}{dx}(uvw)=uv\frac{dw}{dx}+uw\frac{dv}{dx}+vw\frac{du}{dx}$
b. Repeat the above process: $\frac{d}{dx}(u_1u_2u_3u_4)=u_1u_2u_3\frac{du_4}{dx}+u_1u_2u_4\frac{du_3}{dx}+u_1u_3u_4\frac{du_2}{dx}+u_2u_3u_4\frac{du_1}{dx}$
c. The above results can be generalized to $n$ products: $\frac{d}{dx}(u_1u_2u_3...u_n)=u_2u_2u_3...u_n\frac{du_1}{dx}+u_1u_3...u_n\frac{du_2}{dx}+u_1u_2...u_n\frac{du_23}{dx}+...+u_1u_2...u_{n-1}\frac{du_n}{dx}$
