veneciasp

2022-07-14

Need help with logarithmic differentitation
I have the expression
$y=\sqrt{{x}^{2}\left(x+1\right)\left(x+2\right)}.$
I have tried looking at videos but I still cannot arrive at the correct answer and don't know how to get there.
By the way, the correct answer is
${y}^{\prime }=\frac{4{x}^{2}+9x+4}{2\sqrt{\left(x+1\right)\left(x+2\right)}}.$

Ordettyreomqu

Expert

$y=\sqrt{{x}^{2}\left(x+1\right)\left(x+2\right)}$
$\mathrm{ln}y=\mathrm{ln}x+\frac{1}{2}\mathrm{ln}\left(x+1\right)+\frac{1}{2}\mathrm{ln}\left(x+2\right)$
$\frac{{y}^{\prime }}{y}=\frac{1}{x}+\frac{1}{2}\frac{1}{x+1}+\frac{1}{2}\frac{1}{x+2}$
${y}^{\prime }=x\sqrt{\left(x+1\right)\left(x+2\right)}\left(\frac{1}{x}+\frac{1}{2}\frac{1}{x+1}+\frac{1}{2}\frac{1}{x+2}\right)=x\sqrt{\left(x+1\right)\left(x+2\right)}\left(\frac{4{x}^{2}+9x+4}{2x\left(x+1\right)\left(x+2\right)}\right)$