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

1 Answer
Mar 23, 2018

(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))