<s>[INST] Find $f'(0)$ for $f(x)=(2x+1)^3(3x+3)^2$. Find $f'(0)$ for $f(x)=(2x+1)^3(3x+3)^2$.
Do I use the chain rule for each or do I use the derivative product rule first
Please Help!!!
 [/INST] $f(x) = (2x+1)^3\cdot (3x+3)^3$ and $f(0) = 27$
Taking $\log_{e}$ on both side
$\log_{e}f(x) = \log\left\{(2x+1)^3 \cdot (3x+3)^3\right\} = 3\cdot \log_{e}(2x+1)+3\cdot \log_{e}(3x+3)$
Now Diff. both side w.r.to $x$
$\displaystyle \frac{1}{f(x)}\cdot f^{'}(x) = \frac{3}{2x+1}\cdot 2+\frac{3}{3x+3}\cdot 3$
Put $x = 0$ on both side,
$\displaystyle \frac{f^{'}(0)}{27} = \frac{6}{1}+\frac{9}{3} = 9$
So $f^{'}(0) = 27 \times 9 = 243$
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