5th derivative.

I'm assuming that Log (x)   is base 10

This can get a little messy.....here's WolframAlpha's answer........

https://www.wolframalpha.com/input/?i=fifth+derivative++%5Bsin+(x)+*+Log+(x)%5D&rawformassumption=%7B%22FunClash%22,+%22Log%22%7D+-%3E+%7B%22Log10%22%7D

or if it is log base e

$y=e^xsinx\\~\\ y'=e^xsinx+e^xcosx\\~\\ y''=e^xsinx+e^xcosx+e^xcosx-e^xsinx\\ y''=2e^xcosx\\~\\ y'''=2e^xcosx-2e^xsinx\\ y'''=2e^x(cosx-sinx)\\ y''''=2e^x(cosx-sinx)+2e^x(-sinx-cosx)\\ y''''=2e^x(cosx-sinx-sinx-cosx)\\ y''''=-4e^xsinx\\ y'''''=-4y'=-4e^x(sinx+cosx)\\ $

If the log base is "e" ....it's more simple......but still a little messy

5th derivative of   f(x)   =  sin x * ln (x)

 It will be the same answer as before except  that we will not have any logs in the denominators :

d^5/dx^5(sin(x) log(x)) = (24 sin(x))/x^5 - (30 cos(x))/x^4 - (20 sin(x))/x^3 + (10 cos(x))/x^2 + (5 sin(x))/x + log(x) cos(x)

See Wolfram's computation, here :

https://www.wolframalpha.com/input/?i=fifth+derivative+%5B+sin+x+*+ln+(x)+%5D