Differentiate $(x + 1)(x + 2)^2(x + 3)^3$ Obviously we can "brute force" this by multiplying the various terms and differentiating from there. But based upon the solution provided in the text where I found this problem, it looks like there's a "cleaner" method available.
Of note, this problem appears in the chapter introducing the chain rule.
Any ideas?
UPDATE: just to clarify, this problem appears in a very basic calculus text, so any/all "advanced" differentiation techniques would not be at the student's disposal. It's also worth pointing out the answer provided in the text:
$2(x + 2)(x + 3)^2(3x^2 + 11x + 9)$

There is also the logarithmic differentiation approach: http://en.wikipedia.org/wiki/Logarithmic_differentiation
Note that $\ln((x+1)(x+2)^2(x+3)^3)=\ln(x+1)+2\ln(x+2)+3\ln(x+3)$.
Then $\frac{d}{dx}\ln(f(x))=\frac{f^{\prime}(x)}{f(x)}$ using the chain rule.
So take the derivative of the RHS (above) to get
$$\frac{f^\prime(x)}{f(x)} = \frac{1}{x+1}+\frac{2}{x+2}+\frac{3}{x+3}$$
Finish by multiplying both sides of the equation by the original function:
\begin{align*}f^\prime(x) &= f(x)\left(\frac{1}{x+1}+\frac{2}{x+2}+\frac{3}{x+3}\right) \\
&= (x+2)^2(x+3)^3 + 2(x+1)(x+2)(x+3)^3 + 3(x+1)(x+2)^2(x+3)^2
\end{align*}
