Question

What is the derivative of `(ln x)^(1/5)`?

Answer 1

`1/(5x(lnx)^(4/5))`

Explanation:

differentiate using the `color(blue)"chain rule"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(dy/dx=(dy)/(du)xx(du)/(dx))color(white)(a/a)|)))........ (A)`

`color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(d/dx(lnx)=1/x)color(white)(a/a)|)))`

let `y=(lnx)^(1/5)`

now let `u=lnxrArr(du)/(dx)=1/x`

and y `=u^(1/5)rArr(dy)/(du)=1/5u^(-4/5)`

substitute these values into (A) convert u back to x.

`rArrdy/dx=1/5u^(-4/5)xx1/x=1/(5x(lnx)^(4/5)`