What are the first and second derivatives of #f(x)=x^8*ln(x)#?

1 Answer
Guilherme N.
Dec 24, 2015

Let's remember product rule: be #y=f(x)g(x)#, then #y'=f'(x)g(x)+f(x)g'(x)#

Explanation:

Solving:

#(df(x))/(dx)=8x^7lnx+x^8(1/x)=8x^7lnx+x^7=x^7(8lnx+1)#

Using product rule again:

#(df(x))^2/(d^2x)=7x^6(8lnx+1)+x^7(8/x)#

#(df(x))^2/(d^2x)=56x^6lnx+7x^6+8x^6=56x^6lnx+15x^6#

#(df(x))^2/(d^2x)=x^6(56lnx+15)#

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