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

1 Answer
Jim G.
Aug 24, 2016

#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)#

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