How do you find the derivative of #sqrt(x ln(x^4))#?

1 Answer
Mar 23, 2018

Answer:

#(ln(x^4)+4)/(2sqrt(xln(x^4)))#

Explanation:

Let's rewrite it as:
#[(xln(x^4))^(1/2)]'#

Now we have to derivate from the outside to the inside using the chain rule.
#1/2[xln(x^4)]^(-1/2)*[xln(x^4)]'#

Here we got a derivative of a product
#1/2(xln(x^4))^(-1/2)*[(x')ln(x^4)+x(ln(x^4))']#

#1/2(xln(x^4))^(-1/2)*[1*ln(x^4)+x(1/x^4*4x^3)]#

Just using basic algebra to get a semplified version:
#1/2(xln(x^4))^(-1/2)*[ln(x^4)+4]#

And we get the solution:
#(ln(x^4)+4)/(2sqrt(xln(x^4)))#


By the way you can even rewrite the inital problem to make it more simple:
#sqrt(4xln(x))#
#\sqrt(4)sqrt(xln(x))#
#2sqrt(xln(x))#