Question

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

Answer 1

`(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)))`

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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))`