Need help with logarithmic differentiation

Question

I need to use logarithmic differentiation to get f(x)=x$\sqrt{(x+1)(x+2)(x+3)(x+4)}$. I've been working on it for a while and could use some help. Thanks!

Answer 1

$f(x) =x \prod_{k=1}^4 \sqrt{x+k} $.

$\ln f(x) =\ln x + \sum_{k=1}^4 \frac12 \ln(x+k) $.

$\frac{f'(x)}{f(x)} =(\ln f(x))' = \frac1{x} + \sum_{k=1}^4 \frac12 \frac1{x+k} $.

$f'(x) =f(x)(\ln f(x))' =f(x)\left(\frac1{x} + \sum_{k=1}^4 \frac12 \frac1{x+k}\right) =\left(x \prod_{k=1}^4 \sqrt{x+k}\right)\left(\frac1{x} + \sum_{k=1}^4 \frac12 \frac1{x+k}\right) $.

You can distribute the left product over the right sum as you wish.