What is the derivative of #((sin(x))/(2x^4+x))^7#?

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Answer 1 · 2017-05-01T16:44:59

#(df)/(dx)=7(sinx/(2x^4+x))^6xx(2x^4cosx+xcosx-8x^3sinx-sinx)/(2x^4+x)^2#

Here we have #f(x)=(sinx/(2x^4+x))^7#

Let #g(x)=sinx/(2x^4+x)# and then #f(x)=(g(x))^7#

then using quotient rule #(dg)/(dx)=((2x^4+x)cosx-sinx(8x^3+1))/(2x^4+x)^2#

then using chain rule

#(df)/(dx)=(df)/(dg)xx(dg)/(dx)#

= #7(g(x))^6xx((2x^4+x)cosx-sinx(8x^3+1))/(2x^4+x)^2#

= #7(sinx/(2x^4+x))^6xx(2x^4cosx+xcosx-8x^3sinx-sinx)/(2x^4+x)^2#