Derivative of log(y)

Question

I am quite stuck on how to differentiate log(y) with respect to y. When I googled it, I seen many websites state that d(log(y))/dy is 1/yln(10). However, the question I have attached requires me to differentiate xylog(y) w/ respect to y, but I got a different answer to my teacher. I have attached the question, teacher’s answer & my answer. Thanks

Answer 1

You are interpreting the question as using base $10$ while your teacher uses base $e$ for the logarithm.

Hence the difference in your solution.

Your answer is correct if base $10$ is indeed used in the logarithm though the popular default is base $e$ in calculus. Not always, but popular.

Edit:

$$\frac{\partial}{\partial y}(xy \log_b y)=x \log_b y+\frac{xy}{\ln b}$$

You are treating the question as $b=10$ while your teacher is treating $b=e$.