Question

How do you differentiate `f(x)=sqrtcos(lnx^2)` using the chain rule?

Answer 1

found: `=1/x*(-sin(ln(x^2)))/sqrt(cos(ln(x^2))`

Explanation:

Here, basically, you have 4 functions one inside the other! Starting for the most "external" you have:
the square root or `()^(1/2)`, the `cos`, then the `ln` and finally the square `()^2`.
To derive you start from the outer one (the square root) and subsequently multiply by each derivative up the inner one (colored).
So:
given: `f(x)=(cos(ln(x^2)))^(1/2)` you get:

`f'(x)=1/2(cos(ln(x^2)))^(1/2-1)*color(red)(-sin(ln(x^2)))*color(blue)(1/x^2)*color(green)(2x)=`

rearranging:

`=(cos(ln(x^2)))^(-1/2)*(-sin(ln(x^2)))1/x=`

`=1/x*(-sin(ln(x^2)))/sqrt(cos(ln(x^2))`