Chapter 3.4, Problem 98E

a.

To determine

To prove: ${\left(fg\right)}^{\left(2\right)}={f^{\prime }}^{\prime }g+2f^{\prime }{g}^{\prime }+f{g^{\prime }}^{\prime }$.

b.

To determine

To prove: The formula ${\left(fg\right)}^{\left(n\right)}={\displaystyle \sum _{k=0}^n\left(\begin{array}{l}n\\ k\end{array}\right)f^{\left(k\right)}{g}^{\left(n-k\right)}}$ where $\left(\begin{array}{l}n\\ k\end{array}\right)=\frac{n!}{k!\left(n-k\right)!}$ are the binomial coefficients.

c.

To determine

To compare: The result of (b) to the expansion of ${\left(a+b\right)}^n$.