What is the derivative of #sqrt(x ln(x^4))#?

1 Answer

Answer:

The trick to this problem is to break it down into a few steps.

Explanation:

  1. Take the derivative of the square root:
    #sqrt(xln(x^4))=(xln(x^4))^(1/2)#, so the derivative is #1/2(xln(x^4))^(-1/2)=1/(2sqrt(xln(x^4)))#.
  2. Take the derivative of the inside (product rule + chain rule):
    So the derivative is, #(1)ln(x^4)+(x)(1/x^4)(4x^3)=ln(x^4)+4x^4/x^4=ln(x^4)+4#.
  3. Therefore, our total derivative is:
    #1/(2sqrt(xln(x^4)))(ln(x^4)+4)=(ln(x^4)+4)/(2sqrt(xln(x^4))#.