find `dy/dx` using logarithmic differentiation: `x*root(4)(1+x^5)`

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Let, `y=x*root(4)(1+x^5)`

Taking log of both sides,

`lny=lnx+1/4ln(1+x^5)` (note that `ln(uv)=lnu+lnv` , and `lnu^v=vlnu` )

Differentiate with respect to x, note that the left hand side differential and the second differential on the right hand side differential require chain rule,

`(1/y)dy/dx=1/x+1/4*1/(1+x^5)*5x^4`

`=1/x+5/4*1/(1+x^5)*x^4`

`rArr dy/dx=y(1/x+5/4*1/(1+x^5)*x^4)`

`=xroot(4)(1+x^5)(1/x+5/4*1/(1+x^5)*x^4)`

`=root(4)(1+x^5)+5/4*x^5/root(4)((1+x^5)^3)`

=>answer

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