How do you find the derivative of x^(7x)x7x?

1 Answer
Apr 30, 2016

d/dx x^(7x)=7x^(7x)(ln(x)+1)ddxx7x=7x7x(ln(x)+1)

Explanation:

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

  • d/dx e^x = e^xddxex=ex
  • d/dx ln(x) = 1/xddxln(x)=1x
  • d/dx x = 1ddxx=1

we have

d/dx x^(7x) = d/dx e^(ln(x^(7x)))ddxx7x=ddxeln(x7x)

=d/dx e^(7xln(x))=ddxe7xln(x)

=e^(7xln(x))(d/dx7xln(x))=e7xln(x)(ddx7xln(x))

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

=7e^(ln(x^(7x)))(xd/dxln(x) + ln(x)d/dxx)=7eln(x7x)(xddxln(x)+ln(x)ddxx)

(by the product rule, and factoring out the 77)

=7x^(7x)(x*1/x + ln(x)*1)=7x7x(x1x+ln(x)1)

=7x^(7x)(ln(x)+1)=7x7x(ln(x)+1)