How do you find the derivative of #x^(7x)#?

1 Answer
Apr 30, 2016

Answer:

#d/dx x^(7x)=7x^(7x)(ln(x)+1)#

Explanation:

Using the chain rule and the product rule, together with the following derivatives:

  • #d/dx e^x = e^x#
  • #d/dx ln(x) = 1/x#
  • #d/dx x = 1#

we have

#d/dx x^(7x) = d/dx e^(ln(x^(7x)))#

#=d/dx e^(7xln(x))#

#=e^(7xln(x))(d/dx7xln(x))#

(by the chain rule with the functions #e^x# and #7xln(x)#)

#=7e^(ln(x^(7x)))(xd/dxln(x) + ln(x)d/dxx)#

(by the product rule, and factoring out the #7#)

#=7x^(7x)(x*1/x + ln(x)*1)#

#=7x^(7x)(ln(x)+1)#