What are the first and second derivatives of #f(x)=-3x^5lnx #?

1 Answer
Guilherme N.
Dec 24, 2015

Here, we'll need the product rule, which states that for #y=f(x)g(x)#, #y'=f'(x)g(x)+f(x)g'(x)#

Explanation:

Solving:

#(df(x))/(dx)=(-15x^4)(lnx)+(-3x^5)(1/x)=-15x^4lnx-3x^4=color(green)(-3x^4(5lnx+1))#

Second derivative follows the same logic:

#(df(x))^2/(dx)=(-12x^3)(5lnx+1)+(-3x^4)(5/x)=-60x^3lnx-12-15x^3#

#(df(x))^2/(dx)=color(blue)(-15x^3(4lnx+1)-12)#

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