What is the derivative of #f(x) = sin(ln(x+x^2))#?

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Answer 1 · 2018-07-31T07:02:39

#(dy)/(dx)=(1+2x)/(x+x^2)cos(ln(x+x^2)#

Here,

#f(x)=y=sin(ln(x+x^2))#

Let ,

#y=sinu# , #and color(brown)(u=ln(x+x^2)#

#(dy)/(du)=cosu# , #and (du)/(dx)=1/(x+x^2)*(1+2x)=(1+2x)/(x(1+x))#

#color(blue)((dy)/(dx)=(dy)/(du)(du)/(dx)#

#=>(dy)/(dx)=cosu((1+2x)/(x(1+x)))#

Subst. back #color(brown)(u=ln(x+x^2)#

#:.(dy)/(dx)=cos(ln(x+x^2))(1+2x)/(x(1+x))#

#:.(dy)/(dx)=(1+2x)/(x(1+x))cos(ln(x+x^2)#